Running a T-test
Josie Lobring
2024-11-02
1 Loading Libraries
# install any packages you have not previously used, then comment them back out.
#install.packages("car")
#install.packages("effsize")
library(psych) # for the describe() command
library(car) # for the leveneTest() command## Loading required package: carData##
## Attaching package: 'car'## The following object is masked from 'package:psych':
##
## logitlibrary(effsize) # for the cohen.d() command##
## Attaching package: 'effsize'## The following object is masked from 'package:psych':
##
## cohen.d2 Importing Data
d <- read.csv(file="Data/projectdata.csv", header=T)
# For the HW, you will import the project dataset you cleaned previously
# This will be the dataset you'll use for HWs throughout the rest of the semester3 State Your Hypothesis
We predict that there will be a significant difference in depression by age (under 18 versus over 45), as measured by the Patient Health Questionnaire (PHQ-9).
4 Check Your Variables
# you only need to check the variables you're using in the current analysis
## Checking the Categorical variable (IV)
str(d)## 'data.frame': 917 obs. of 7 variables:
## $ X : int 20 30 31 33 57 58 81 104 113 117 ...
## $ age : chr "1 under 18" "1 under 18" "4 between 36 and 45" "4 between 36 and 45" ...
## $ urban_rural: chr "city" "city" "town" "city" ...
## $ big5_con : num 3.33 5.33 5.67 6 3.33 ...
## $ phq : num 3.33 1 2.33 1.11 2.33 ...
## $ support : num 2.17 5 2.5 3.67 4.17 ...
## $ swemws : num 2.29 4.29 3.29 4 3.29 ...# if the categorical variable you're using is showing as a "chr" (character), you must change it to be a factor -- using the next line of code (as.factor)
d$age <- as.factor(d$age)
table(d$age, useNA = "always")##
## 1 under 18 2 between 18 and 25 3 between 26 and 35 4 between 36 and 45
## 599 53 6 86
## 5 over 45 <NA>
## 173 0## Checking the Continuous variable (DV)
# you can use the describe() command on an entire dataframe (d) or just on a single variable within your dataframe -- which we will do here
describe(d$phq)## vars n mean sd median trimmed mad min max range skew kurtosis se
## X1 1 917 2.05 0.85 1.89 1.97 0.99 1 4 3 0.66 -0.61 0.03# also use a histogram to visualize your continuous variable
hist(d$phq)
# use the describeBy() command to view the means and standard deviations by group
# it's very similar to the describe() command but splits the dataframe according to the 'group' variable
describeBy(d$phq, group=d$age)##
## Descriptive statistics by group
## group: 1 under 18
## vars n mean sd median trimmed mad min max range skew kurtosis se
## X1 1 599 2.25 0.86 2.11 2.2 0.99 1 4 3 0.4 -0.87 0.04
## ------------------------------------------------------------
## group: 2 between 18 and 25
## vars n mean sd median trimmed mad min max range skew kurtosis se
## X1 1 53 2.52 0.78 2.44 2.52 0.99 1 4 3 0 -1.05 0.11
## ------------------------------------------------------------
## group: 3 between 26 and 35
## vars n mean sd median trimmed mad min max range skew kurtosis se
## X1 1 6 1.81 0.21 1.83 1.81 0.16 1.44 2 0.56 -0.71 -1.07 0.08
## ------------------------------------------------------------
## group: 4 between 36 and 45
## vars n mean sd median trimmed mad min max range skew kurtosis se
## X1 1 86 1.63 0.65 1.33 1.52 0.33 1 3.78 2.78 1.59 2 0.07
## ------------------------------------------------------------
## group: 5 over 45
## vars n mean sd median trimmed mad min max range skew kurtosis se
## X1 1 173 1.43 0.47 1.22 1.35 0.33 1 3.56 2.56 1.56 2.83 0.04# last, use a boxplot to examine your continuous and categorical variables together
boxplot(d$phq~d$age)
5 Check Your Assumptions
5.1 T-test Assumptions
- IV must have two levels
- Data values must be independent (independent t-test only)
- Data obtained via a random sample
- Dependent variable must be normally distributed
- Variances of the two groups are approximately equal
# If the IV has more than 2 levels, you must drop the additional levels so that you meet the first assumption of a t-test.
d <- subset(d, age != "4 between 36 and 45")
table(d$age, useNA = "always") #verify that now there are no participants in the removed level##
## 1 under 18 2 between 18 and 25 3 between 26 and 35 4 between 36 and 45
## 599 53 6 0
## 5 over 45 <NA>
## 173 0d <- subset(d, age != "3 between 26 and 35")
table(d$age, useNA = "always") #verify that now there are no participants in the removed level##
## 1 under 18 2 between 18 and 25 3 between 26 and 35 4 between 36 and 45
## 599 53 0 0
## 5 over 45 <NA>
## 173 0d <- subset(d, age != "2 between 18 and 25")
table(d$age, useNA = "always") #verify that now there are no participants in the removed level##
## 1 under 18 2 between 18 and 25 3 between 26 and 35 4 between 36 and 45
## 599 0 0 0
## 5 over 45 <NA>
## 173 0d$age <- droplevels(d$age) # use droplevels() to drop the empty factor
table(d$age, useNA = "always") #verify that now the entire factor level is removed ##
## 1 under 18 5 over 45 <NA>
## 599 173 05.2 Testing Homogeneity of Variance with Levene’s Test
We can test whether the variances of our two groups are equal using Levene’s test. The null hypothesis is that the variance between the two groups is equal, which is the result we want. So when running Levene’s test we’re hoping for a non-significant result!
# use the leveneTest() command from the car package to test homogeneity of variance
# it uses the same 'formula' setup that we'll use for our t-test: formula is y~x, where y is our DV and x is our IV
leveneTest(phq~age, data = d)## Levene's Test for Homogeneity of Variance (center = median)
## Df F value Pr(>F)
## group 1 90.536 < 2.2e-16 ***
## 770
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1As you can see, the data has significantly different variances between the two comparison groups.
When running a t-test, we can account for heterogeneity in our variance by using Welch’s t-test, which does not have the same assumption about variance as Student’s t-test (the general default type of t-test). R defaults to using Welch’s t-test so this doesn’t require any changes on our part! Even if your data has no issues with homogeneity of variance, you’ll still use Welch’s t-test – it handles the potential issues around variance well and there are no real downsides. We’re just using Levene’s test here to get into the habit of checking the homogeneity of our variance, even if we already have the solution for any potential problems.
5.3 Issues with My Data
My independent variable has more than two levels. To proceed with this analysis, I will drop the participants between the ages of 18 and 25, 26 and 35, and 36 and 45 from my sample. I will make a note to discuss this issue in my Method write-up and in my Discussion as a limitation of my study.
My data also has some issues regarding homogeneity of variance as Levene’s test was significant. To accommodate for this heterogeneity of variance, I will use Welch’s t-test instead of Student’s t-test in my analysis.
6 Run a T-test
# very simple! we use the same formula of y~x, where y is our DV and x is our IV
t_output <- t.test(d$phq~d$age)7 View Test Output
t_output##
## Welch Two Sample t-test
##
## data: d$phq by d$age
## t = 16.502, df = 526.17, p-value < 2.2e-16
## alternative hypothesis: true difference in means between group 1 under 18 and group 5 over 45 is not equal to 0
## 95 percent confidence interval:
## 0.7254630 0.9215359
## sample estimates:
## mean in group 1 under 18 mean in group 5 over 45
## 2.250603 1.4271038 Calculate Cohen’s d
# once again, we use the same formula, y~x, to calculate cohen's d
d_output <- cohen.d(d$phq~d$age)9 View Effect Size
d_output##
## Cohen's d
##
## d estimate: 1.045801 (large)
## 95 percent confidence interval:
## lower upper
## 0.8684931 1.223108210 Write Up Results
To test our hypothesis that there will be a significant difference in depression by age (under 18 versus over 45), we used an independent samples t-test. This required us to drop our participants between the ages of 18 and 45 from our sample, as we are limited to a two-group comparison when using this test. We tested the homogeneity of variance with Levene’s test and found signs of heterogeneity (p < .001). This suggests that there is an increased chance of Type I error. To correct for this issue, we used Welch’s t-test, which does not assume homogeneity of variance. Our data met all other assumptions of a t-test.
Our hypothesis was supported, specifically we found that participants under the age of 18 (M = 2.25, SD = 0.86) reported significantly higher levels of depression than participants over the age of 45 (M = 1.43, SD = 0.47); t(526.17) = 16.502, p < .001 (see Figure 1). The effect size was calculated using Cohen’s d, with a value of 1.05 (large effect; Cohen, 1988).
References
Cohen J. (1988). Statistical Power Analysis for the Behavioral Sciences. New York, NY: Routledge Academic.