1. Intro

As a data scientist, you’ll frequently have to deal with messy and high-dimensional datasets. In this chapter, you’ll learn how to use Principal Component Analysis (PCA) to effectively reduce the dimensionality of such datasets so that it becomes easier to extract actionable insights from them.

(1) Why the high dimension bother?

  • Big Computatinoal cost to handle high-dimensional data.
  • Estimation accuracy decreases.
  • Difficult interpretation of the data.

(2) How do we trace correlation patterns?

  • Correlation matrix is a matrix of correlation coefficients.
  • Smaller number of dimensions translate to less complex correlation matrix.

(3) How do we deal with the Curse of Dimensionality? Two Solutions:

  • Feature Engineering: Requires domain knowledge.
  • Remove redundancy.

(4) Why reduce dimensionality?

It is to explain as much data variation as possible while discarding highly correlated variables.

2. Exploring multivariate data

We’ve loaded a data frame called cars into your workspace. Go ahead and explore it! It includes features of a big range of brands of cars from 2004. In this exercise, you will explore the dataset and attempt to draw useful conclusions from the correlation matrix. Recall that correlation reveals feature resemblance and it will help us infer how cars are related to each other based on their features’ values. To this end, you will discover how difficult it is to trace patterns based solely on the correlation structure.

(1) Explore cars with the summary() function and take a first look at the complexity of the data.

data("mtcars")
mtcars$cyl <- as.numeric(as.character(mtcars$cyl))

(2) Create a correlation matrix of the columns 9 to 19 of cars with cor(), using only complete rows (i.e, “complete.obs”). For the next exercises, it’s worth keeping in mind that these numeric columns 9-19 are relevant to implementing PCA.

# Get the correlation matrix with cor()
correl <- cor(mtcars, use = "complete.obs")

(3) Use the ggcorrplot() function of the ggcorrplot package for a better overview of the correl object.

library(ggcorrplot)
## Loading required package: ggplot2
ggcorrplot(correl)

(4) You can also use ggcorrplot() to conduct hierarchical clustering on correlation matrices.

# Conduct hierarchical clustering on the correlation matrix
ggcorrplot_clustered <- ggcorrplot(correl, hc.order = TRUE, type = "lower")
ggcorrplot_clustered

3. PCA: What does it do?

  1. Conceptually:
  • Removes correlation.
  • Extracts new dimenstions(=principal components)
  • Reveals the true dimensionality of the data
  1. Practically:
  • Decomposes the correlation matrix
  • Changes the coordinate system.
  • Helps reduce the number of dimensions.

(1) The five steps to Perform

  1. Pre-Prcocessing steps
  • Centering
  • Standardisation
  1. Change of coordinate system
  • Rotation
  • Projection
  1. Explained Variance
  • Reduction

4. PCA with FactoMineR

As you saw in the video, FactoMineR is a very useful package, rich in functionality, that implements a number of dimensionality reduction methods. Its function for doing PCA is PCA() - easy to remember! Recall that PCA(), by default, generates 2 graphs and extracts the first 5 PCs. You can use the ncp argument to manually set the number of dimensions to keep.

You can also use the summary() function to get a quick overview of the indices of the first three principal components. Moreover, for extracting summaries of some of the rows in a dataset, you can specify the nbelements argument. You’ll have a chance to practice all of these and more in this exercise!

As in the previous lesson, the cars dataset is available in your workspace.

(1) Run PCA() on the 10 numeric variables (9 to 19) of cars and extract 4 dimensions. To set the number of dimensions, you need to use the ncp argument of PCA(). The graph argument is set to F to suppress the default graphical output PCA().

library(FactoMineR)
# Run a PCA for the 10 non-binary numeric variables of cars
pca_output_ten_v <- PCA(mtcars[, 2:11], ncp = 4, graph = FALSE)

(2) Browse through the trimmed version of the first 100 cars in the PCA model object pca_output_ten_v. For the trimmed version, you need the nbelements argument of the summary() function.

# Get the summary of the first 10 cars
summary(pca_output_ten_v, nbelements = 10)
## 
## Call:
## PCA(X = mtcars[, 2:11], ncp = 4, graph = FALSE) 
## 
## 
## Eigenvalues
##                        Dim.1   Dim.2   Dim.3   Dim.4   Dim.5   Dim.6
## Variance               5.760   2.650   0.597   0.270   0.222   0.210
## % of var.             57.602  26.496   5.972   2.695   2.223   2.101
## Cumulative % of var.  57.602  84.099  90.071  92.766  94.988  97.089
##                        Dim.7   Dim.8   Dim.9  Dim.10
## Variance               0.133   0.081   0.054   0.024
## % of var.              1.329   0.807   0.537   0.238
## Cumulative % of var.  98.419  99.226  99.762 100.000
## 
## Individuals (the 10 first)
##                       Dist    Dim.1    ctr   cos2    Dim.2    ctr   cos2  
## Mazda RX4         |  2.228 | -0.632  0.217  0.080 |  1.740  3.570  0.610 |
## Mazda RX4 Wag     |  2.075 | -0.605  0.199  0.085 |  1.554  2.849  0.561 |
## Datsun 710        |  2.952 | -2.802  4.258  0.901 | -0.123  0.018  0.002 |
## Hornet 4 Drive    |  2.511 | -0.259  0.036  0.011 | -2.364  6.593  0.887 |
## Hornet Sportabout |  2.445 |  2.033  2.241  0.691 | -0.775  0.708  0.100 |
## Valiant           |  2.995 | -0.205  0.023  0.005 | -2.779  9.107  0.861 |
## Duster 360        |  3.034 |  2.846  4.395  0.880 |  0.318  0.119  0.011 |
## Merc 240D         |  2.746 | -1.939  2.039  0.498 | -1.454  2.494  0.280 |
## Merc 230          |  3.705 | -2.300  2.871  0.385 | -1.964  4.547  0.281 |
## Merc 280          |  1.901 | -0.637  0.220  0.112 | -0.151  0.027  0.006 |
##                    Dim.3    ctr   cos2  
## Mazda RX4         -0.665  2.315  0.089 |
## Mazda RX4 Wag     -0.435  0.988  0.044 |
## Datsun 710        -0.415  0.899  0.020 |
## Hornet 4 Drive    -0.095  0.047  0.001 |
## Hornet Sportabout -1.016  5.405  0.173 |
## Valiant            0.093  0.046  0.001 |
## Duster 360        -0.324  0.550  0.011 |
## Merc 240D          0.956  4.779  0.121 |
## Merc 230           1.751 16.047  0.223 |
## Merc 280           1.434 10.761  0.569 |
## 
## Variables
##                      Dim.1    ctr   cos2    Dim.2    ctr   cos2    Dim.3
## cyl               |  0.967 16.239  0.935 |  0.064  0.152  0.004 | -0.107
## disp              |  0.950 15.676  0.903 | -0.088  0.291  0.008 | -0.013
## hp                |  0.850 12.555  0.723 |  0.399  6.001  0.159 |  0.141
## drat              | -0.757  9.960  0.574 |  0.453  7.755  0.205 |  0.101
## wt                |  0.880 13.454  0.775 | -0.239  2.154  0.057 |  0.298
## qsec              | -0.528  4.836  0.279 | -0.750 21.221  0.562 |  0.311
## vs                | -0.800 11.113  0.640 | -0.370  5.177  0.137 |  0.319
## am                | -0.594  6.126  0.353 |  0.703 18.663  0.495 | -0.182
## gear              | -0.531  4.903  0.282 |  0.757 21.638  0.573 |  0.216
## carb              |  0.544  5.140  0.296 |  0.670 16.949  0.449 |  0.434
##                      ctr   cos2  
## cyl                1.925  0.011 |
## disp               0.027  0.000 |
## hp                 3.322  0.020 |
## drat               1.705  0.010 |
## wt                14.884  0.089 |
## qsec              16.247  0.097 |
## vs                17.017  0.102 |
## am                 5.520  0.033 |
## gear               7.800  0.047 |
## carb              31.553  0.188 |

(3) Fill in the missing part of the code for extracting the variance of the first three newly-extracted dimensions by accessing the eig attribute of pca_output_ten_v. Notice that pca_output_ten_v’s eig attribute represents a dataframe.

# Get the variance of the first 3 new dimensions.
pca_output_ten_v$eig[,2][1:3]
##    comp 1    comp 2    comp 3 
## 57.602174 26.496432  5.972149

(4) After completing the exercise, make sure to explore the resulting PCA model object and the variance matrices in the console.

# Get the cumulative variance.
pca_output_ten_v$eig[,3][1:3]
##   comp 1   comp 2   comp 3 
## 57.60217 84.09861 90.07075

5. Exploring PCA()

PCA() provides great flexibility in its usage. You can choose to ignore some of the original variables or individuals in building a PCA model by supplying PCA() with the ind.sup argument for supplementary individuals and quanti.sup or quali.sup for quantitative and qualitative variables respectively. Supplementary individuals and variables are rows and variables of the original data ignored while building the model.

Your learning objectives in this exercise are:

  • To conduct PCA considering parts of a dataset
  • To inspect the most correlated variables with a specified principal component
  • To find the contribution of variables in the designation of the first 5 principal components

(1) Run a PCA on cars by setting the first 7 variables as quantitative supplementary variables and the last 4 variables as qualitative supplementary ones. The graph argument is set to F for not displaying the default graphical output.

# Run a PCA with active and supplementary variables
library(dplyr)
## 
## Attaching package: 'dplyr'
## The following objects are masked from 'package:stats':
## 
##     filter, lag
## The following objects are masked from 'package:base':
## 
##     intersect, setdiff, setequal, union
mtcars[, 8:11] <- mtcars[, 8:11] %>% 
  mutate_if(is.factor, as.character)
str(mtcars)
## 'data.frame':    32 obs. of  11 variables:
##  $ mpg : num  21 21 22.8 21.4 18.7 18.1 14.3 24.4 22.8 19.2 ...
##  $ cyl : num  6 6 4 6 8 6 8 4 4 6 ...
##  $ disp: num  160 160 108 258 360 ...
##  $ hp  : num  110 110 93 110 175 105 245 62 95 123 ...
##  $ drat: num  3.9 3.9 3.85 3.08 3.15 2.76 3.21 3.69 3.92 3.92 ...
##  $ wt  : num  2.62 2.88 2.32 3.21 3.44 ...
##  $ qsec: num  16.5 17 18.6 19.4 17 ...
##  $ vs  : num  0 0 1 1 0 1 0 1 1 1 ...
##  $ am  : num  1 1 1 0 0 0 0 0 0 0 ...
##  $ gear: num  4 4 4 3 3 3 3 4 4 4 ...
##  $ carb: num  4 4 1 1 2 1 4 2 2 4 ...
pca_output_all <- PCA(mtcars, quali.sup = 8:11, graph = FALSE)

(2) Use dimdesc() on the pca_output_all object to inspect the most correlated variables with the first two principal components by specifying the axes argument.

# Get the most correlated variables
dimdesc(pca_output_all, axes = 1:2)
## $Dim.1
## $Dim.1$quanti
##      correlation      p.value
## cyl    0.9578708 8.370479e-18
## disp   0.9528463 4.386676e-17
## wt     0.8825092 2.406474e-11
## hp     0.8744936 6.125817e-11
## carb   0.5603408 8.518999e-04
## gear  -0.4877022 4.634276e-03
## qsec  -0.5410937 1.384932e-03
## am    -0.5611453 8.342340e-04
## drat  -0.7468681 9.129050e-07
## vs    -0.7718009 2.314162e-07
## mpg   -0.9308661 1.176137e-14
## 
## 
## $Dim.2
## $Dim.2$quanti
##      correlation      p.value
## qsec   0.8058477 2.623038e-08
## hp    -0.3623835 4.152388e-02
## drat  -0.4824288 5.168925e-03
## carb  -0.4918156 4.250884e-03
## gear  -0.6191451 1.581008e-04
## am    -0.6422701 7.402925e-05

(3) Perform the same PCA as previously, but this time only on the first 100 car categories (rows). Consider the rest rows/individuals, namely from the 101st until the last one (use nrow() for the total number of rows) as supplementary ones.

# Run a PCA on the first 10 car categories
pca_output_ten <- PCA(mtcars, quanti.sup = 1:5, quali.sup = 8:11, ind.sup = 11:nrow(mtcars), graph=FALSE)

(4) Check out the contributions of the variables on the first 5 principal components by exploring the var property of the pca_output_hundred object.

# Trace variable contributions in pca_output_hundred
pca_output_ten$var$contrib
##      Dim.1 Dim.2
## wt      50    50
## qsec    50    50

You have used the most important arguments of the main PCA() function, namely quali.sup, quanti.sup and ind.sup. Moreover, you have been browsing through the available atrributes of the resulting PCA model object!

6. PCA with ade4

Alright! Now that you’ve got some real hands-on experience with FactoMineR, let’s have a look at ade4, a well-known and well-maintained R package with a large number of numerical methods for building and handling PCA models. dudi.pca() is the main function that implements PCA for ade4 and by default, it is interactive: It lets the user insert the number of retained dimensions. For suppressing the interactive mode and inserting the number of axes within the dudi.pca() function, you need to set the scannf argument to FALSE and then use the nf argument for setting the number of axes to retain. So, let’s put ade4 into practice and compare it with FactoMineR.

# Run a PCA using the 4 non-binary numeric variables.
library(ade4)
## 
## Attaching package: 'ade4'
## The following object is masked from 'package:FactoMineR':
## 
##     reconst
mtcars[, 8:11] <- mtcars[, 8:11] %>% 
  mutate_if(is.character, as.factor)
str(mtcars)
## 'data.frame':    32 obs. of  11 variables:
##  $ mpg : num  21 21 22.8 21.4 18.7 18.1 14.3 24.4 22.8 19.2 ...
##  $ cyl : num  6 6 4 6 8 6 8 4 4 6 ...
##  $ disp: num  160 160 108 258 360 ...
##  $ hp  : num  110 110 93 110 175 105 245 62 95 123 ...
##  $ drat: num  3.9 3.9 3.85 3.08 3.15 2.76 3.21 3.69 3.92 3.92 ...
##  $ wt  : num  2.62 2.88 2.32 3.21 3.44 ...
##  $ qsec: num  16.5 17 18.6 19.4 17 ...
##  $ vs  : num  0 0 1 1 0 1 0 1 1 1 ...
##  $ am  : num  1 1 1 0 0 0 0 0 0 0 ...
##  $ gear: num  4 4 4 3 3 3 3 4 4 4 ...
##  $ carb: num  4 4 1 1 2 1 4 2 2 4 ...
mtcars_pca <- dudi.pca(mtcars[,1:7], scannf = FALSE, nf = 4)

# Explore the summary of cars_pca.
summary(mtcars_pca)
## Class: pca dudi
## Call: dudi.pca(df = mtcars[, 1:7], scannf = FALSE, nf = 4)
## 
## Total inertia: 7
## 
## Eigenvalues:
##     Ax1     Ax2     Ax3     Ax4     Ax5 
##  5.0861  1.1566  0.3449  0.1579  0.1295 
## 
## Projected inertia (%):
##     Ax1     Ax2     Ax3     Ax4     Ax5 
##  72.659  16.522   4.926   2.256   1.850 
## 
## Cumulative projected inertia (%):
##     Ax1   Ax1:2   Ax1:3   Ax1:4   Ax1:5 
##   72.66   89.18   94.11   96.36   98.21 
## 
## (Only 5 dimensions (out of 7) are shown)
# Explore the summary of pca_output_ten_v.
summary(pca_output_ten_v)
## 
## Call:
## PCA(X = mtcars[, 2:11], ncp = 4, graph = FALSE) 
## 
## 
## Eigenvalues
##                        Dim.1   Dim.2   Dim.3   Dim.4   Dim.5   Dim.6
## Variance               5.760   2.650   0.597   0.270   0.222   0.210
## % of var.             57.602  26.496   5.972   2.695   2.223   2.101
## Cumulative % of var.  57.602  84.099  90.071  92.766  94.988  97.089
##                        Dim.7   Dim.8   Dim.9  Dim.10
## Variance               0.133   0.081   0.054   0.024
## % of var.              1.329   0.807   0.537   0.238
## Cumulative % of var.  98.419  99.226  99.762 100.000
## 
## Individuals (the 10 first)
##                       Dist    Dim.1    ctr   cos2    Dim.2    ctr   cos2  
## Mazda RX4         |  2.228 | -0.632  0.217  0.080 |  1.740  3.570  0.610 |
## Mazda RX4 Wag     |  2.075 | -0.605  0.199  0.085 |  1.554  2.849  0.561 |
## Datsun 710        |  2.952 | -2.802  4.258  0.901 | -0.123  0.018  0.002 |
## Hornet 4 Drive    |  2.511 | -0.259  0.036  0.011 | -2.364  6.593  0.887 |
## Hornet Sportabout |  2.445 |  2.033  2.241  0.691 | -0.775  0.708  0.100 |
## Valiant           |  2.995 | -0.205  0.023  0.005 | -2.779  9.107  0.861 |
## Duster 360        |  3.034 |  2.846  4.395  0.880 |  0.318  0.119  0.011 |
## Merc 240D         |  2.746 | -1.939  2.039  0.498 | -1.454  2.494  0.280 |
## Merc 230          |  3.705 | -2.300  2.871  0.385 | -1.964  4.547  0.281 |
## Merc 280          |  1.901 | -0.637  0.220  0.112 | -0.151  0.027  0.006 |
##                    Dim.3    ctr   cos2  
## Mazda RX4         -0.665  2.315  0.089 |
## Mazda RX4 Wag     -0.435  0.988  0.044 |
## Datsun 710        -0.415  0.899  0.020 |
## Hornet 4 Drive    -0.095  0.047  0.001 |
## Hornet Sportabout -1.016  5.405  0.173 |
## Valiant            0.093  0.046  0.001 |
## Duster 360        -0.324  0.550  0.011 |
## Merc 240D          0.956  4.779  0.121 |
## Merc 230           1.751 16.047  0.223 |
## Merc 280           1.434 10.761  0.569 |
## 
## Variables
##                      Dim.1    ctr   cos2    Dim.2    ctr   cos2    Dim.3
## cyl               |  0.967 16.239  0.935 |  0.064  0.152  0.004 | -0.107
## disp              |  0.950 15.676  0.903 | -0.088  0.291  0.008 | -0.013
## hp                |  0.850 12.555  0.723 |  0.399  6.001  0.159 |  0.141
## drat              | -0.757  9.960  0.574 |  0.453  7.755  0.205 |  0.101
## wt                |  0.880 13.454  0.775 | -0.239  2.154  0.057 |  0.298
## qsec              | -0.528  4.836  0.279 | -0.750 21.221  0.562 |  0.311
## vs                | -0.800 11.113  0.640 | -0.370  5.177  0.137 |  0.319
## am                | -0.594  6.126  0.353 |  0.703 18.663  0.495 | -0.182
## gear              | -0.531  4.903  0.282 |  0.757 21.638  0.573 |  0.216
## carb              |  0.544  5.140  0.296 |  0.670 16.949  0.449 |  0.434
##                      ctr   cos2  
## cyl                1.925  0.011 |
## disp               0.027  0.000 |
## hp                 3.322  0.020 |
## drat               1.705  0.010 |
## wt                14.884  0.089 |
## qsec              16.247  0.097 |
## vs                17.017  0.102 |
## am                 5.520  0.033 |
## gear               7.800  0.047 |
## carb              31.553  0.188 |

Now, you know how to conduct PCA with the dudi.pca() function of ade4, another well-known package for conducting dimensionality reduction.

7. Plotting cos2

You’re getting the hang of PCA now! As Alex demonstrated in the video, an important index included in your PCA models is the squared cosine, abbreviated in FactoMineR and factoextra as cos2. This shows how accurate the representation of your variables or individuals on the PC plane is.

The factoextra package is excellent at handling PCA models built using FactoMineR. Here, you’re going to explore the functionality of factoextra. You’ll be using the pca_output_all object that you computed earlier and create plots based on its cos2. Visual aids are key to understanding cos2.

(1) Create a factor map using fviz_pca_var() for the variables with cos2 higher than 0.7.

library(factoextra)
## Welcome! Related Books: `Practical Guide To Cluster Analysis in R` at https://goo.gl/13EFCZ
fviz_pca_var(pca_output_all, select.var = list(cos2 = 0.7), repel = TRUE)

(2) Modify your code to create the individuals’ factor map using fviz_pca_ind() and select.ind.

# Modify the code to create a factor map for the individuals.
fviz_pca_ind(pca_output_all, select.ind = list(cos2 = 0.7), repel = TRUE)

(3) Create a barplot for the 10 variables with the highest cos2 on the 1st PC. Use “var” for variables, instead of “ind”.

# Create a barplot for the variables with the highest cos2 in the 1st PC.
fviz_cos2(pca_output_all, choice = "var", axes = 1, top = 10)

(4) Modify your code to create a barplot for the variables with the highest cos2 in the 2nd PC.

# Create a barplot for the variables with the highest cos2 in the 2nd PC.
fviz_cos2(pca_output_all, choice = "var", axes = 2, top = 10)

Take a minute and think: What do the cos2 tell you about the new representation provided by your models? Can you tell whether they are good enough in quality?

8. Plotting contributions

In this exercise, you will be asked to prepare a number of plots to help you get a better feeling of the variables’ contributions on the extracted principal components. It is important to keep in mind that the contributions of the variables essentially signify their importance for the construction of a given principal component.

(1) Create a plot of the top 5 variables with the highest contributions.

# Create a factor map for the top 5 variables with the highest contributions.
fviz_pca_var(pca_output_all, select.var = list(contrib = 5), repel = TRUE)

(2) Create a plot of the top 5 individuals with the highest contributions. For this, you will need to create the individuals’ factor map.

fviz_pca_ind(pca_output_all, select.ind = list(contrib = 5), repel = TRUE)

(3) Time to put fviz_contrib() into practice. Create a barplot of the 5 variables with the highest contributions on the 1st principal component. Then, create the exact same type of barplot but this time on the 2nd principal component.

# Create a barplot for the variables with the highest contributions to the 1st PC.
fviz_contrib(pca_output_all, choice = "var", axes = 1, top = 5)

# Create a barplot for the variables with the highest contributions to the 2nd PC.
fviz_contrib(pca_output_all, choice = "var", axes = 2, top = 5)

You know what it means that a variable is important in the extraction of the principal components! You could compare the two different graphs. Which variables are in both dimensions very important and which ones are only in one of them?

9. Biplots and their ellipsoids

As mentioned in the video, biplots are graphs that provide a compact way of summarizing the relationships between individuals, variables, and also between variables and individuals within the same plot. Moreover, ellipsoids can be added on top of a biplot and offer a much better overview of the biplot based on the groupings of variables and individuals.

In this exercise, your job is to create biplots and ellipsoids using factoextra’s graphical utilities.

(1) Create a biplot of pca_output_all that helps visualise all individuals and variables in the same plot. Notice the directions of the arrows with respect to the point clouds and try to interpret the displayed biplot.

# Create a biplot with no labels for all individuals with the geom argument.
fviz_pca_biplot(pca_output_all)

(2) Ellipsoids can be added within the individuals factors’ map to group individuals that resemble each other in the coordinate system of the principal components. Create ellipsoids based on the levels of the supplementary variable am

# Create ellipsoids with habillage for am
fviz_pca_ind(pca_output_all, habillage = mtcars$am, addEllipses = TRUE)
## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse

(3) Create the biplot with ellipsoids in it based on the levels of wheeltype, and display the quality of representation of the variables accordingly. For this, you need to set the alpha.var argument of the required function to “cos2”.

# Create the biplot with ellipsoids
fviz_pca_biplot(pca_output_all, habillage = mtcars$am, addEllipses = TRUE, alpha.var = "cos2")
## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced

## Warning in stats::qt(ci/2 + 0.5, data_sum$length - 1): NaNs produced
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse
## Too few points to calculate an ellipse

Excellent! You have used visual means for presenting the results of dimensionality reduction! You are now almost ready to move on to more advanced material on PCA.

10. Proximity of variables in a 2-D PCA plot

What does it mean when two variables are close to each other in a biplot?

Select the option below that provides the correct answer to this question. Which variables are very important in both dimensions and which are only important in one of them?

Possible Answers? 1. Both variables are close in distance and close to the radius of the unit correlation circle. 2. Both variables are close in distance with lots of observations. 3. Both variables have small Euclidean distance.

The answer is 3. It’s about time to dig into choices you have to make when conducting PCA. In the next chapter, you’ll start by using some formal criteria in deciding how many PCs to retain.