Introduction to linear regression
Juan Falck
The Human Freedom Index is a report that attempts to summarize the idea of “freedom” through a bunch of different variables for many countries around the globe. It serves as a rough objective measure for the relationships between the different types of freedom - whether it’s political, religious, economical or personal freedom - and other social and economic circumstances. The Human Freedom Index is an annually co-published report by the Cato Institute, the Fraser Institute, and the Liberales Institut at the Friedrich Naumann Foundation for Freedom.
In this lab, you’ll be analyzing data from Human Freedom Index reports from 2008-2016. Your aim will be to summarize a few of the relationships within the data both graphically and numerically in order to find which variables can help tell a story about freedom.
Getting Started
Load packages
In this lab, you will explore and visualize the data using the tidyverse suite of packages. The data can be found in the companion package for OpenIntro resources, openintro.
Let’s load the packages.
library(tidyverse)
library(openintro)
rm(list=ls())
data('hfi', package='openintro')The data
The data we’re working with is in the openintro package and it’s called hfi, short for Human Freedom Index.
- What are the dimensions of the dataset?
ANSWER
dim(hfi)## [1] 1458 123The table is 1,458 rows with 123 columns
- What type of plot would you use to display the relationship between the personal freedom score,
pf_score, and one of the other numerical variables? Plot this relationship using the variablepf_expression_controlas the predictor. Does the relationship look linear? If you knew a country’spf_expression_control, or its score out of 10, with 0 being the most, of political pressures and controls on media content, would you be comfortable using a linear model to predict the personal freedom score?
ANSWER
ggplot(hfi, aes(x=pf_expression_control, y=pf_score)) +
geom_point(alpha=0.5) +
geom_smooth(method=lm, se=FALSE)
We do see a linear relationship between pf_expression_control and the resulting pf_score.
If the relationship looks linear, we can quantify the strength of the relationship with the correlation coefficient.
hfi %>%
summarise(cor(pf_expression_control, pf_score, use = "complete.obs"))## # A tibble: 1 x 1
## `cor(pf_expression_control, pf_score, use = "complete.obs")`
## <dbl>
## 1 0.796Here, we set the use argument to “complete.obs” since there are some observations of NA.
Sum of squared residuals
In this section, you will use an interactive function to investigate what we mean by “sum of squared residuals”. You will need to run this function in your console, not in your markdown document. Running the function also requires that the hfi dataset is loaded in your environment.
Think back to the way that we described the distribution of a single variable. Recall that we discussed characteristics such as center, spread, and shape. It’s also useful to be able to describe the relationship of two numerical variables, such as pf_expression_control and pf_score above.
- Looking at your plot from the previous exercise, describe the relationship between these two variables. Make sure to discuss the form, direction, and strength of the relationship as well as any unusual observations.
ANSWER We do see an positive relationship between pf_expression_control and pf_score, the variance seems constant (even some reduction) from low pf_expression_control scores to the highest one. We do see some outliers mainly in the both extremes of pf_expression_control.
Just as you’ve used the mean and standard deviation to summarize a single variable, you can summarize the relationship between these two variables by finding the line that best follows their association. Use the following interactive function to select the line that you think does the best job of going through the cloud of points.
# This will only work interactively (i.e. will not show in the knitted document)
hfi <- hfi %>% filter(complete.cases(pf_expression_control, pf_score))
DATA606::plot_ss(x = hfi$pf_expression_control, y = hfi$pf_score)After running this command, you’ll be prompted to click two points on the plot to define a line. Once you’ve done that, the line you specified will be shown in black and the residuals in blue. Note that there are 30 residuals, one for each of the 30 observations. Recall that the residuals are the difference between the observed values and the values predicted by the line:
\[ e_i = y_i - \hat{y}_i \]
The most common way to do linear regression is to select the line that minimizes the sum of squared residuals. To visualize the squared residuals, you can rerun the plot command and add the argument showSquares = TRUE.
DATA606::plot_ss(x = hfi$pf_expression_control, y = hfi$pf_score, showSquares = TRUE)Note that the output from the plot_ss function provides you with the slope and intercept of your line as well as the sum of squares.
- Using
plot_ss, choose a line that does a good job of minimizing the sum of squares. Run the function several times. What was the smallest sum of squares that you got? How does it compare to your neighbors?
ANSWER The smallest Sum of Squares I was able to obtain was 93,530
Call: lm(formula = y ~ x, data = pts)
Coefficients: (Intercept) x
13.1556 0.4261
Sum of Squares: 93530.03
The linear model
It is rather cumbersome to try to get the correct least squares line, i.e. the line that minimizes the sum of squared residuals, through trial and error. Instead, you can use the lm function in R to fit the linear model (a.k.a. regression line).
m1 <- lm(pf_score ~ pf_expression_control, data = hfi)The first argument in the function lm is a formula that takes the form y ~ x. Here it can be read that we want to make a linear model of pf_score as a function of pf_expression_control. The second argument specifies that R should look in the hfi data frame to find the two variables.
The output of lm is an object that contains all of the information we need about the linear model that was just fit. We can access this information using the summary function.
summary(m1)##
## Call:
## lm(formula = pf_score ~ pf_expression_control, data = hfi)
##
## Residuals:
## Min 1Q Median 3Q Max
## -3.8467 -0.5704 0.1452 0.6066 3.2060
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 4.61707 0.05745 80.36 <2e-16 ***
## pf_expression_control 0.49143 0.01006 48.85 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.8318 on 1376 degrees of freedom
## (80 observations deleted due to missingness)
## Multiple R-squared: 0.6342, Adjusted R-squared: 0.634
## F-statistic: 2386 on 1 and 1376 DF, p-value: < 2.2e-16Let’s consider this output piece by piece. First, the formula used to describe the model is shown at the top. After the formula you find the five-number summary of the residuals. The “Coefficients” table shown next is key; its first column displays the linear model’s y-intercept and the coefficient of at_bats. With this table, we can write down the least squares regression line for the linear model:
\[ \hat{y} = 4.61707 + 0.49143 \times pf\_expression\_control \]
One last piece of information we will discuss from the summary output is the Multiple R-squared, or more simply, \(R^2\). The \(R^2\) value represents the proportion of variability in the response variable that is explained by the explanatory variable. For this model, 63.42% of the variability in runs is explained by at-bats.
- Fit a new model that uses
pf_expression_controlto predicthf_score, or the total human freedom score. Using the estimates from the R output, write the equation of the regression line. What does the slope tell us in the context of the relationship between human freedom and the amount of political pressure on media content?
ANSWER
m2 <- lm(hf_score ~ pf_expression_control, data = hfi)
summary(m2)##
## Call:
## lm(formula = hf_score ~ pf_expression_control, data = hfi)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2.6198 -0.4908 0.1031 0.4703 2.2933
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 5.153687 0.046070 111.87 <2e-16 ***
## pf_expression_control 0.349862 0.008067 43.37 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.667 on 1376 degrees of freedom
## (80 observations deleted due to missingness)
## Multiple R-squared: 0.5775, Adjusted R-squared: 0.5772
## F-statistic: 1881 on 1 and 1376 DF, p-value: < 2.2e-16The regression equation is:
\[ \hat{y} = 5.153687 + 0.349862 \times pf\_expression\_control \] Again we see a positive corelation between the response variable hf_score and the predictor pf_expression_control
Prediction and prediction errors
Let’s create a scatterplot with the least squares line for m1 laid on top.
ggplot(data = hfi, aes(x = pf_expression_control, y = pf_score)) +
geom_point() +
stat_smooth(method = "lm", se = FALSE)
Here, we are literally adding a layer on top of our plot. geom_smooth creates the line by fitting a linear model. It can also show us the standard error se associated with our line, but we’ll suppress that for now.
This line can be used to predict \(y\) at any value of \(x\). When predictions are made for values of \(x\) that are beyond the range of the observed data, it is referred to as extrapolation and is not usually recommended. However, predictions made within the range of the data are more reliable. They’re also used to compute the residuals.
- If someone saw the least squares regression line and not the actual data, how would they predict a country’s personal freedom school for one with a 6.7 rating for
pf_expression_control? Is this an overestimate or an underestimate, and by how much? In other words, what is the residual for this prediction?
ANSWER
By inspecting only the regression line we would estimate that a pf_expression_control = 6.7 would yield me a pf_score of ~8 (a little less to be exact).
hfi %>%
select(pf_expression_control,pf_score) %>%
filter(pf_expression_control>=6.5 & pf_expression_control<6.8)## # A tibble: 79 x 2
## pf_expression_control pf_score
## <dbl> <dbl>
## 1 6.75 7.43
## 2 6.75 8.22
## 3 6.75 8.77
## 4 6.75 7.87
## 5 6.5 7.54
## 6 6.5 8.77
## 7 6.75 7.39
## 8 6.75 7.25
## 9 6.5 8.35
## 10 6.5 8.76
## # ... with 69 more rowsWe can see that there are no values for pf_expression_control = 6.7. Scores in that ranges where either 6.75 of 6.5. But if we check the scores = 6.75 (closest to 6.7) I would say the prediction is pretty much in the center of the actual observed values.
Model diagnostics
To assess whether the linear model is reliable, we need to check for (1) linearity, (2) nearly normal residuals, and (3) constant variability.
Linearity: You already checked if the relationship between pf_score and `pf_expression_control’ is linear using a scatterplot. We should also verify this condition with a plot of the residuals vs. fitted (predicted) values.
ggplot(data = m1, aes(x = .fitted, y = .resid)) +
geom_point() +
geom_hline(yintercept = 0, linetype = "dashed") +
xlab("Fitted values") +
ylab("Residuals")
Notice here that m1 can also serve as a data set because stored within it are the fitted values (\(\hat{y}\)) and the residuals. Also note that we’re getting fancy with the code here. After creating the scatterplot on the first layer (first line of code), we overlay a horizontal dashed line at \(y = 0\) (to help us check whether residuals are distributed around 0), and we also reanme the axis labels to be more informative.
- Is there any apparent pattern in the residuals plot? What does this indicate about the linearity of the relationship between the two variables?
ANSWER I don’t see a clear pattern in the plot of fitted vs residual. This implies that they are independent.
Nearly normal residuals: To check this condition, we can look at a histogram
ggplot(data = m1, aes(x = .resid)) +
geom_histogram(binwidth = 0.25) +
xlab("Residuals")
or a normal probability plot of the residuals.
ggplot(data = m1, aes(sample = .resid)) +
stat_qq()
Note that the syntax for making a normal probability plot is a bit different than what you’re used to seeing: we set sample equal to the residuals instead of x, and we set a statistical method qq, which stands for “quantile-quantile”, another name commonly used for normal probability plots.
- Based on the histogram and the normal probability plot, does the nearly normal residuals condition appear to be met?
ANSWER The histogram of residuals looks fairly “normal” and the qq-plot is pretty linear for most of the sample. So yes, I believe criteria is met.
Constant variability:
- Based on the residuals vs. fitted plot, does the constant variability condition appear to be met?
ANSWER The plot doesn’t show changes in the range of the residuals, so YES, constant variability seems to also be met.
More Practice
- Choose another freedom variable and a variable you think would strongly correlate with it.. Produce a scatterplot of the two variables and fit a linear model. At a glance, does there seem to be a linear relationship?
ANSWER Let’s check the relationship of pf_score and pf_association_assembly
ggplot(data = hfi, aes(x = pf_association_assembly, y = pf_score)) +
geom_point() +
stat_smooth(method = "lm", se = FALSE)
m3 <- lm(pf_score ~ pf_association_assembly, data = hfi)
summary(m3)##
## Call:
## lm(formula = pf_score ~ pf_association_assembly, data = hfi)
##
## Residuals:
## Min 1Q Median 3Q Max
## -4.1728 -0.6902 0.1041 0.7784 2.5704
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 4.72658 0.07903 59.81 <2e-16 ***
## pf_association_assembly 0.34163 0.00991 34.47 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.9747 on 1127 degrees of freedom
## (329 observations deleted due to missingness)
## Multiple R-squared: 0.5133, Adjusted R-squared: 0.5128
## F-statistic: 1188 on 1 and 1127 DF, p-value: < 2.2e-16There seems to be a strong linear relatioship between pf_score and pf_association_assembly
- How does this relationship compare to the relationship between
pf_expression_controlandpf_score? Use the \(R^2\) values from the two model summaries to compare. Does your independent variable seem to predict your dependent one better? Why or why not?
ANSWER The R-squared of pf_association_assemnly is 0.51, not as high as the 0.63 of pf_expression_control but still high enough to be a good predictor of pf_score
summary(m1)$r.squared## [1] 0.6342361summary(m3)$r.squared## [1] 0.5132725- What’s one freedom relationship you were most surprised about and why? Display the model diagnostics for the regression model analyzing this relationship.
ANSWER Let’s generate a corelation matrix and see how variables corelate to pf_score.
my_cor_table <- hfi %>%
select(pf_rol_procedural:hf_quartile) %>%
cor(use = "pairwise.complete.obs")
#Isolate pf_score
pf_cor <- my_cor_table["pf_score",]#We could use as.data.frame as well
df <- as_tibble(as.list(my_cor_table[,"pf_score"]))
df## # A tibble: 1 x 119
## pf_rol_procedural pf_rol_civil pf_rol_criminal pf_rol pf_ss_homicide
## <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 0.877 0.693 0.682 0.774 0.244
## # ... with 114 more variables: pf_ss_disappearances_disap <dbl>,
## # pf_ss_disappearances_violent <dbl>, pf_ss_disappearances_organized <dbl>,
## # pf_ss_disappearances_fatalities <dbl>, pf_ss_disappearances_injuries <dbl>,
## # pf_ss_disappearances <dbl>, pf_ss_women_fgm <dbl>,
## # pf_ss_women_missing <dbl>, pf_ss_women_inheritance_widows <dbl>,
## # pf_ss_women_inheritance_daughters <dbl>, pf_ss_women_inheritance <dbl>,
## # pf_ss_women <dbl>, pf_ss <dbl>, pf_movement_domestic <dbl>, ...#List all column names in the list
names(df)## [1] "pf_rol_procedural" "pf_rol_civil"
## [3] "pf_rol_criminal" "pf_rol"
## [5] "pf_ss_homicide" "pf_ss_disappearances_disap"
## [7] "pf_ss_disappearances_violent" "pf_ss_disappearances_organized"
## [9] "pf_ss_disappearances_fatalities" "pf_ss_disappearances_injuries"
## [11] "pf_ss_disappearances" "pf_ss_women_fgm"
## [13] "pf_ss_women_missing" "pf_ss_women_inheritance_widows"
## [15] "pf_ss_women_inheritance_daughters" "pf_ss_women_inheritance"
## [17] "pf_ss_women" "pf_ss"
## [19] "pf_movement_domestic" "pf_movement_foreign"
## [21] "pf_movement_women" "pf_movement"
## [23] "pf_religion_estop_establish" "pf_religion_estop_operate"
## [25] "pf_religion_estop" "pf_religion_harassment"
## [27] "pf_religion_restrictions" "pf_religion"
## [29] "pf_association_association" "pf_association_assembly"
## [31] "pf_association_political_establish" "pf_association_political_operate"
## [33] "pf_association_political" "pf_association_prof_establish"
## [35] "pf_association_prof_operate" "pf_association_prof"
## [37] "pf_association_sport_establish" "pf_association_sport_operate"
## [39] "pf_association_sport" "pf_association"
## [41] "pf_expression_killed" "pf_expression_jailed"
## [43] "pf_expression_influence" "pf_expression_control"
## [45] "pf_expression_cable" "pf_expression_newspapers"
## [47] "pf_expression_internet" "pf_expression"
## [49] "pf_identity_legal" "pf_identity_parental_marriage"
## [51] "pf_identity_parental_divorce" "pf_identity_parental"
## [53] "pf_identity_sex_male" "pf_identity_sex_female"
## [55] "pf_identity_sex" "pf_identity_divorce"
## [57] "pf_identity" "pf_score"
## [59] "pf_rank" "ef_government_consumption"
## [61] "ef_government_transfers" "ef_government_enterprises"
## [63] "ef_government_tax_income" "ef_government_tax_payroll"
## [65] "ef_government_tax" "ef_government"
## [67] "ef_legal_judicial" "ef_legal_courts"
## [69] "ef_legal_protection" "ef_legal_military"
## [71] "ef_legal_integrity" "ef_legal_enforcement"
## [73] "ef_legal_restrictions" "ef_legal_police"
## [75] "ef_legal_crime" "ef_legal_gender"
## [77] "ef_legal" "ef_money_growth"
## [79] "ef_money_sd" "ef_money_inflation"
## [81] "ef_money_currency" "ef_money"
## [83] "ef_trade_tariffs_revenue" "ef_trade_tariffs_mean"
## [85] "ef_trade_tariffs_sd" "ef_trade_tariffs"
## [87] "ef_trade_regulatory_nontariff" "ef_trade_regulatory_compliance"
## [89] "ef_trade_regulatory" "ef_trade_black"
## [91] "ef_trade_movement_foreign" "ef_trade_movement_capital"
## [93] "ef_trade_movement_visit" "ef_trade_movement"
## [95] "ef_trade" "ef_regulation_credit_ownership"
## [97] "ef_regulation_credit_private" "ef_regulation_credit_interest"
## [99] "ef_regulation_credit" "ef_regulation_labor_minwage"
## [101] "ef_regulation_labor_firing" "ef_regulation_labor_bargain"
## [103] "ef_regulation_labor_hours" "ef_regulation_labor_dismissal"
## [105] "ef_regulation_labor_conscription" "ef_regulation_labor"
## [107] "ef_regulation_business_adm" "ef_regulation_business_bureaucracy"
## [109] "ef_regulation_business_start" "ef_regulation_business_bribes"
## [111] "ef_regulation_business_licensing" "ef_regulation_business_compliance"
## [113] "ef_regulation_business" "ef_regulation"
## [115] "ef_score" "ef_rank"
## [117] "hf_score" "hf_rank"
## [119] "hf_quartile"df2 <- as_tibble(cbind(nms = names(df), t(df)))
df2## # A tibble: 119 x 2
## nms V2
## <chr> <chr>
## 1 pf_rol_procedural 0.877298271073169
## 2 pf_rol_civil 0.693163766056389
## 3 pf_rol_criminal 0.682376550363886
## 4 pf_rol 0.774387600506464
## 5 pf_ss_homicide 0.243847139820907
## 6 pf_ss_disappearances_disap 0.535361401608651
## 7 pf_ss_disappearances_violent 0.373247536151383
## 8 pf_ss_disappearances_organized 0.657469551536047
## 9 pf_ss_disappearances_fatalities 0.386863728009172
## 10 pf_ss_disappearances_injuries 0.3505752691219
## # ... with 109 more rowsWe can inspect all corelations by looking at pf_cor. The one variable that I found surprising was pf_religion_restrictions.
ggplot(data = hfi, aes(x = pf_religion_restrictions, y = pf_score)) +
geom_point() +
stat_smooth(method = "lm", se = FALSE)
m4 <- lm(pf_score ~ pf_religion_restrictions, data = hfi)
summary(m4)##
## Call:
## lm(formula = pf_score ~ pf_religion_restrictions, data = hfi)
##
## Residuals:
## Min 1Q Median 3Q Max
## -4.2226 -0.9581 -0.1356 1.1526 2.5061
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 6.0503 0.1559 38.798 < 2e-16 ***
## pf_religion_restrictions 0.1617 0.0208 7.772 1.51e-14 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.314 on 1362 degrees of freedom
## (94 observations deleted due to missingness)
## Multiple R-squared: 0.04247, Adjusted R-squared: 0.04177
## F-statistic: 60.41 on 1 and 1362 DF, p-value: 1.507e-14summary(m1)$r.squared## [1] 0.6342361summary(m3)$r.squared## [1] 0.5132725summary(m4)$r.squared## [1] 0.04247127I would’ve though that pf_religion_assembly would’ve been a good predictor of pf_score, but with with a R-Squared = 0.0424 it doesn’t seem to be a good predictor for pf_score