Descriptive Statistics

A study by Miller et al. (2004) compared the survival of two kinds of Lake Superior rainbow trout fry (babies). Four thousand fry were from a government hatchery on the lake, whereas 4000 more fry came from wild trout. All 8000 fry were released into a stream flowing into the lake, where they remained for one year. After one year, the researchers found 78 survivors. Of these, 27 were hatchery fish and 51 were wild. Display these results in the most appropriate table. Identify the type of table you used.

A study by Miller et al. (2004) compared the survival of two kinds of Lake Superior rainbow trout fry (babies). Four thousand fry were from a government hatchery on the lake, whereas 4000 more fry came from wild trout. All 8000 fry were released into a stream flowing into the lake, where they remained for one year. After one year, the researchers found 78 survivors. Of these, 27 were hatchery fish and 51 were wild. Display these results in the most appropriate table. Identify the type of table you used.

Question: Experimental or observational?

Answer: Observational

A study by Miller et al. (2004) compared the survival of two kinds of Lake Superior rainbow trout fry (babies). Four thousand fry were from a government hatchery on the lake, whereas 4000 more fry came from wild trout. All 8000 fry were released into a stream flowing into the lake, where they remained for one year. After one year, the researchers found 78 survivors. Of these, 27 were hatchery fish and 51 were wild. Display these results in the most appropriate table. Identify the type of table you used.

Question: Explanatory variables with type?

Answer: Fry source, which is categorical variable with two levels, “hatchery” and “wild”.

A study by Miller et al. (2004) compared the survival of two kinds of Lake Superior rainbow trout fry (babies). Four thousand fry were from a government hatchery on the lake, whereas 4000 more fry came from wild trout. All 8000 fry were released into a stream flowing into the lake, where they remained for one year. After one year, the researchers found 78 survivors. Of these, 27 were hatchery fish and 51 were wild. Display these results in the most appropriate table. Identify the type of table you used.

Question: Response variables with type?

Answer: “Survival”, which is a categorical variable with two levels, “caught” and “not caught”.

Load the data:

troutfry <- read.csv(paste0(here::here(), "/Datasets/chapter02/chap02q05FrySurvival.csv"))

head(troutfry)

  frySource survival
1      wild survived
2      wild survived
3      wild survived
4      wild survived
5      wild survived
6      wild survived

str(troutfry)

'data.frame':   8000 obs. of  2 variables:
 $ frySource: chr  "wild" "wild" "wild" "wild" ...
 $ survival : chr  "survived" "survived" "survived" "survived" ...

Looks like our data is in raw form, i.e. each row is an observation and each column is a measurement/variable.

(troutfryTable <- table(troutfry$frySource, troutfry$survival))

           not caught survived
  hatchery       3973       27
  wild           3949       51

Notice the parenthesis around the assignment, which says to output the result.

Question: Anything off with this format?

(troutfryTable <- table(troutfry$frySource, troutfry$survival))

           not caught survived
  hatchery       3973       27
  wild           3949       51

Notice the parenthesis around the assignment, which says to output the result.

Answer: Explanatory variable should be in the horizontal dimension!

(troutfryTable <- table(troutfry$frySource, troutfry$survival))

           not caught survived
  hatchery       3973       27
  wild           3949       51

Notice the parenthesis around the assignment, which says to output the result.

(troutfryTable <- table(troutfry$survival, troutfry$frySource))

             hatchery wild
  not caught     3973 3949
  survived         27   51

Question: Any other changes?

(troutfryTable <- table(troutfry$survival, troutfry$frySource))

             hatchery wild
  not caught     3973 3949
  survived         27   51

Answer: Maybe “survived” should come before “not caught”, as that might be the most interesting.

(troutfryTable <- table(troutfry$survival, troutfry$frySource))

             hatchery wild
  not caught     3973 3949
  survived         27   51

Question: How do we change the order of the levels for “survival”?

(troutfryTable <- table(troutfry$survival, troutfry$frySource))

             hatchery wild
  survived         27   51
  not caught     3973 3949

addmargins(troutfryTable)

             hatchery wild  Sum
  survived         27   51   78
  not caught     3973 3949 7922
  Sum            4000 4000 8000

There's more than one way to shear a sheep…

t(addmargins(table(troutfry)))

            frySource
survival     hatchery wild  Sum
  survived         27   51   78
  not caught     3973 3949 7922
  Sum            4000 4000 8000

The t command transposes the table (i.e. switch horizontal and vertical variable placement).

Question: Explanatory variable along vertical axis. How to fix?

Answer: Transpose the table!

mosaicplot(t(troutfryTable), 
           xlab="Fry source", 
           ylab="Relative frequency", 
           main="", 
           cex = 1.5, 
           cex.sub = 1.5, 
           col = c("forestgreen", "goldenrod1"))

The cutoff birth date for school entry in British Columbia, Canada, is December 31. As a result, children born in December tend to be the youngest in their grade, whereas those born in January tend to be the oldest. Morrow et al. (2012) examined how this relative age difference influenced diagnosis and treatment of attention deficit/hyperactivity disorder (ADHD). A total of 39,136 boys aged 6 to 12 years and registered in school in 1997 - 1998 had January birth dates. Of these, 2219 were diagnosed with ADHD in that year. A total of 38,977 boys had December birth dates, of which 2870 were diagnosed with ADHD in that year. Display the association between birth month and ADHD diagnosis using a table or graphical method from this chapter. Is there an association?

The cutoff birth date for school entry in British Columbia, Canada, is December 31. As a result, children born in December tend to be the youngest in their grade, whereas those born in January tend to be the oldest. Morrow et al. (2012) examined how this relative age difference influenced diagnosis and treatment of attention deficit/hyperactivity disorder (ADHD). A total of 39,136 boys aged 6 to 12 years and registered in school in 1997 - 1998 had January birth dates. Of these, 2219 were diagnosed with ADHD in that year. A total of 38,977 boys had December birth dates, of which 2870 were diagnosed with ADHD in that year. Display the association between birth month and ADHD diagnosis using a table or graphical method from this chapter. Is there an association?

Question: Experimental or observational?

Answer: Observational

The cutoff birth date for school entry in British Columbia, Canada, is December 31. As a result, children born in December tend to be the youngest in their grade, whereas those born in January tend to be the oldest. Morrow et al. (2012) examined how this relative age difference influenced diagnosis and treatment of attention deficit/hyperactivity disorder (ADHD). A total of 39,136 boys aged 6 to 12 years and registered in school in 1997 - 1998 had January birth dates. Of these, 2219 were diagnosed with ADHD in that year. A total of 38,977 boys had December birth dates, of which 2870 were diagnosed with ADHD in that year. Display the association between birth month and ADHD diagnosis using a table or graphical method from this chapter. Is there an association?

Question: Explanatory variables with type?

Answer: Birth month - categorical with levels “Jan” and “Dec”.

The cutoff birth date for school entry in British Columbia, Canada, is December 31. As a result, children born in December tend to be the youngest in their grade, whereas those born in January tend to be the oldest. Morrow et al. (2012) examined how this relative age difference influenced diagnosis and treatment of attention deficit/hyperactivity disorder (ADHD). A total of 39,136 boys aged 6 to 12 years and registered in school in 1997 - 1998 had January birth dates. Of these, 2219 were diagnosed with ADHD in that year. A total of 38,977 boys had December birth dates, of which 2870 were diagnosed with ADHD in that year. Display the association between birth month and ADHD diagnosis using a table or graphical method from this chapter. Is there an association?

Question: Response variables with type?

Answer: ADHD diagnosis - categorical with levels “ADHD” and “no ADHD”.

adhd <- read.csv(paste0(here::here(), "/Datasets/chapter02/chap02q33BirthMonthADHD.csv"))
str(adhd)

'data.frame':   4 obs. of  3 variables:
 $ birthMonth : chr  "January" "January" "December" "December"
 $ diagnosis  : chr  "ADHD" "no ADHD" "ADHD" "no ADHD"
 $ frequencies: int  2219 36917 2870 36107

  birthMonth diagnosis frequencies
1    January      ADHD        2219
2    January   no ADHD       36917
3   December      ADHD        2870
4   December   no ADHD       36107

Answer: Processed! They have already counted the data!

In R: Transform this into a more appropriate form.

Look at Chapter 2 R-Markdown, Example 2.3A. Bird malaria to see how to plot with data of this type. In particular, note that grouped barplot (using the barplot command) and mosaic plots (using the mosaicplot command) can work on matrices or tables.

The following data are from Mattison et al. (2012), who carried out an experiment with rhesus monkeys to test whether a reduction in food intake extends life span (as measured in years). The data are the life spans of 19 male and 15 female monkeys who were randomly assigned a normal nutritious diet or a similar diet reduced in amount by 30%. All monkeys were adults at the start of the study.

1. Graph the results, using the most appropriate method and following the four principles of good graph design.
2. According to your graph, which difference in life span is greater: that between the sexes, or that between diet groups?

Other questions: Experimental or observational? Explanatory and response variables?

The following data are from Mattison et al. (2012), who carried out an experiment with rhesus monkeys to test whether a reduction in food intake extends life span (as measured in years). The data are the life spans of 19 male and 15 female monkeys who were randomly assigned a normal nutritious diet or a similar diet reduced in amount by 30%. All monkeys were adults at the start of the study.

Answer: This is an experimental procedure.

Answer: Explanatory variables are sex and food intake, both of which are categorical and have 2 levels.

Answer: Response variable is life span, which is numerical and continuous.

'data.frame':   34 obs. of  3 variables:
 $ sex          : chr  "female" "female" "female" "female" ...
 $ foodTreatment: chr  "reduced" "reduced" "reduced" "reduced" ...
 $ lifespan     : num  16.5 18.9 22.6 27.8 30.2 30.7 35.9 23.7 24.5 24.7 ...

Categories of the sex variable:

[1] "female" "male"  

Categories of the foodTreatment variable:

[1] "control" "reduced"

head(foodforlife)

     sex foodTreatment lifespan
1 female       reduced     16.5
2 female       reduced     18.9
3 female       reduced     22.6
4 female       reduced     27.8
5 female       reduced     30.2
6 female       reduced     30.7

Question: Is this data in the right form for graphing?

Depending on the graph that you choose, the first two arguments to the function should be

functionname(lifespan ~ foodTreatment * sex, data=mydata, …)

where functionname is the name of the plotting function in R that you chose, and mydata is the name of your data frame.

You can combine two columns into one column using the unite function in the tidyr package (we'll learn more later about this package).

foodforlife <- tidyr::unite(foodforlife, category, c("sex", "foodTreatment"), sep="-")
str(foodforlife)

'data.frame':   34 obs. of  2 variables:
 $ category: chr  "female-reduced" "female-reduced" "female-reduced" "female-reduced" ...
 $ lifespan: num  16.5 18.9 22.6 27.8 30.2 30.7 35.9 23.7 24.5 24.7 ...

The values of the new category variable are:

[1] "female-control" "female-reduced" "male-control"   "male-reduced"  

Definition:The frequency distribution of a variable is the number of occurrences of all values of that variable in the data.

Definition:The relative frequency distribution of a variable is the fraction of occurrences of all values of that variable in the data or population.

• These definitions apply to both continuous and discrete variables.
• Frequency = Number
• Relative frequency = Fraction (proportion)

Question:What type of plot represents the frequency (relative frequency) distribution for a discrete variable?

Answer:Bar plot

Definition: A bar plot uses the height of rectangular bars to display the frequency distribution (or relative frequency distribution) of a categorical variable.

• i.e. height of bars = number or proportion

Question: What type of plot represents the frequency distribution for a continuous variable?

Answer: Histogram (which is still a bar plot, actually)

Definition: A histogram for a frequency distribution uses the height of rectangular bars to display the frequency distribution of a numerical variable.

Definition: A histogram for a relative frequency distribution uses the area of rectangular bars to display the relative frequency distribution of a numerical variable.

Load and show the data:

salmonSizeData <- read.csv("http://whitlockschluter.zoology.ubc.ca/wp-content/data/chapter02/chap02f2_5SalmonBodySize.csv")
head(salmonSizeData)

  year   sex oceanAgeYears lengthMm massKg
1 1996 FALSE             3      513  3.090
2 1996 FALSE             3      513  2.909
3 1996 FALSE             3      525  3.056
4 1996 FALSE             3      501  2.690
5 1996 FALSE             3      513  2.876
6 1996 FALSE             3      501  2.978

Plot in a histogram:

histObj <- hist(salmonSizeData$massKg, 
                right = FALSE, 
                breaks = seq(1,4,by=0.5), 
                col = "firebrick")

seq(1,4,by=0.5)

[1] 1.0 1.5 2.0 2.5 3.0 3.5 4.0

Question: What happens with smaller bin width (say width of 0.1)?

hist(salmonSizeData$massKg, 
     right = FALSE, 
     breaks = seq(1,4,by=0.1), 
     col = "firebrick", 
     freq=FALSE)

Definition: The population mean \( \mu \) is the sum of all the observations in the population divided by \( N \), the number of observations in the population (assuming it is finite - for now).
\[ \mu = \frac{1}{N}\sum_{i=1}^{N}Y_{i}\, \]

Definition: The sample mean \( \overline{Y} \) is the sum of all the observations in the sample divided by \( n \), the number of sample observations.
\[ \overline{Y} = \frac{1}{n}\sum_{i=1}^{n}Y_{i}\, \]

Question: Is the population mean \( \mu \) a parameter or an estimate? What about the sample mean?

Note that every observation has equal weight (i.e. \( \frac{1}{n} \)), so any outliers can strongly affect the mean. It is a very democratic statistic - equal representation!

Definition: The population median is the middle measurement of the set of all observations in the population (again, assume population finite for now).

Definition: The sample median is the middle measurement of the set of all observations in the sample.

How do you compute the median? W&S version:

• First, sort the data from smallest to largest.
• We then have two conditions:
  • If the number of observations is odd, then we have \[ Median = Y_{(n+1)/2} \]
  • If the number of observations is even, then we have \[ Median = \left[Y_{n/2} + Y_{(n/2)+1}\right]/2 \]

Look at special cases of \( n=3 \) and \( n=4 \)!!!

The median is the middle measurement of the distibution (different colors represent the two halves of the distribution). The mean is the center of gravity, the point at which the frequency distribution would be balanced (if observations had weight).

Note: The mean and median have the same units as the variable!!!

Definition: The population variance \( \sigma^{2} \) is the average of the squared deviations of all observations from the population mean, and assuming a finite population, we have
\[ \sigma^{2} = \frac{1}{N}\sum_{i=1}^{N}(Y_{i}-\mu)^2 \]

Definition: The sample variance \( s^{2} \) is the average of the squared deviations from the sample mean,
\[ s^{2} = \frac{1}{n-1}\sum_{i=1}^{n}(Y_{i}-\overline{Y})^2 \]

Question: Why \( n-1 \)??

Answer: Needed to be unbiased estimate!!

Definition: The population standard deviation \( \sigma \) is the square root of population variance
\[ \sigma = \sqrt{\sigma^{2}} \]

Definition: The sample standard deviation \( s \) is the square root of the sample variance,
\[ s = \sqrt{s^{2}} \]

Note #1: \( s \) is in general a biased estimator of \( \sigma \). The bias gets smaller as the sample size gets larger.

Note #2: \( s \) and \( \sigma \) have the same units as the random variable!!!

Note #3: If the frequency distribution is bell shaped, then about two-thirds (67%) of the observations will lie within one standard deviation of the mean, and 95% of the observations will lie within two standard deviations of the mean.

Note #3: If the frequency distribution is bell shaped, then about two-thirds (67%) of the observations will lie within one standard deviation of the mean, and 95% of the observations will lie within two standard deviations of the mean.

Definition: The interquartile range \( IQR \) is the difference between the third and first quartiles of the data. It is the span of the middle 50% of the data.

Heuristic #1: The location (mean and median) and spread (interquartile range and standard deviation) give similar information when the frequency distribution is symmetric and unimodal (i.e. bell shaped).

Heuristic #2: The mean and standard deviation become less informative when the distribution is strongly skewed or there there are extreme observations.

Since in biology many times the standard deviation scales with the mean, it can be more informative to look at the coefficient of variation.

Definition: The coefficient of variation (CV) calculates the standard deviation as a percentage of the mean: \[ CV = \frac{s}{\bar{Y}}\times 100\% \]

In other words, the CV answers the question “How much variation is there relative to the mean?”

Make sure you read the book for the following discussions

• How to compute a mean and standard deviation from a frequency table

Question: Why is this important to know?

• Rounding rules for displaying tables and statistics
• Effect of changing measurement scale
• Cumulative frequency distributions (we will cover this later as well)

My point here is that you are responsible for all book material, even if we don't cover it in lecture!

\( ^* \) Note that IQR has different algorithms. To match the algorithm in W&S, you should use IQR(___, type=5). There are different algorithms as there are different ways to calculate quantiles. (for curious souls, see ?quantiles). For the HW, either version is acceptable. Default type in R is type=7.