Content

Before we begin…

These are the packages we will be using in this workbook. If you have been following along with the content each week, this package should already be installed on your computer. Otherwise, make sure you install them using install.packages() if you have not used them before.

library(tidyverse)

Also, something to keep in mind, throughout the next few demonstrations, whenever we analyse data, we will roughly be following this procedure:

Prepare the data for analysis.
Run the statistical test.
Visualise the data.
Write-up analysis.

The General Linear Model in R

R has built in functions that performs all the statistical tests we will be covering this week. Whenever we run an analysis in R (whether it is a t-test, ANOVA, correlation, regression, etc.), generally, we need to provide the function with two things:

The formula we wish to test (more on this below).
The data.frame that holds the data.

All functions take the following template:

function_name(formula,data)

In the next few weeks, we will be covering examples of different analyses. In almost all cases, the function to conduct these analyses follows this template:

Formulas

Whenever we conduct an analysis in R, we need to write the formula for the linear model. In general, this takes the following form:

dependent_variable ~ independent_variable1 + independent_variable2 + ...

The variable on the left of the ~ symbol is the dependent variable you are trying to predict, while variables on the right are your independent variables. The variable names are the same as the column names in your data.frame. Each independent variable in your model (if there is more than one) is separated by the + symbol.

Independent-samples t-test

As covered in the lecture series, an independent samples t-test is used to assess the relationship between a categorical IV (with two levels that is between-subjects) and a continuous DV. In this example, we test whether people who self-identify as a ‘cat-person’ are more introverted than those who self-identify as a ‘dog-person’. This is an ‘independent-samples’ t-test because participants in one group are different to participants in the other group.

1. Prepare the data for analysis.

Click here for details on the Introversion Scale. To keep things simple, we have only used 5 items from this scale. Note: as specified in the link, low scores on this scale indicate higher levels of introversion. Therefore, to be consistent with the name of the measure, scores have been reversed in the dataset (i.e., high scores corresponds with high introversion). Also note: sometimes you will need to reverse code the items yourself, while we don’t have to do this for this data, I have left the code in (commented out) that would reverse score an Item 1 - to do this, we use the recode() function within the mutate() function.

data1.clean <- data %>%
  filter(cat.dog != "both") %>%
  filter(cat.dog != "neither") %>%
  filter(cat.dog != "") %>%
  mutate( #introversion1R = recode(introversion1,`1` = 5,`2` = 4,`3`=3,`4` = 2,`5` = 1),
          introvert = introversion2 + introversion5 + introversion7 + introversion8 + introversion10) %>%
  select(cat.dog,introvert)

In the code above, we have removed participants that did not specify identify as either a “dog-person” or “cat-person”. We have also used the mutate() function to calculate an overall introversion score. We have also only selected the variables we need for our analysis - this step is not strictly necessary, but can help you organise your data.

2. Run the statistical test.

To run the t-test, we use the function t.test(). As mentioned above, the first argument is the formula. In this analysis, the DV is called introvert and the IV is called cat.dog, so the formula is introvert ~ cat.dog. The second argument or the t.test() function is the cleaned data.frame, which we’ve called data1.clean.

t.test(introvert ~ cat.dog,data1.clean)

## 
##  Welch Two Sample t-test
## 
## data:  introvert by cat.dog
## t = 0.25707, df = 29.246, p-value = 0.7989
## alternative hypothesis: true difference in means between group cat and group dog is not equal to 0
## 95 percent confidence interval:
##  -1.901682  2.448691
## sample estimates:
## mean in group cat mean in group dog 
##          13.38462          13.11111

The output from the statistical test gives you a lot of information, but there are a couple bits that are particularly important:

The t-statistic - this is the value for the test statistic (above this equals 0.26)
The degrees of freedom (above this equals 29.25)
The p-value - this tells you whether the test is significant or not (above this equals 0.799)

Can you see where each of these values come from in the output above?

3. Visualise the data.

Box Plot

Box plots are good for representing the range and quantiles of your data. To create a box plot, use geom_boxplot(). Make sure to set the correct variables on the x- and y-axes (cat.dog and introvert respectively). We also map the ‘fill’ aesthetic to the cat.dog variable so that the colours are different.

ggplot(data1.clean,aes(x = cat.dog,y = introvert,fill = cat.dog)) +
  geom_boxplot() +
  theme_classic()

Violin Plot

Violin plots are good for representing the distributions of scores on the DV between groups, and can be created using geom_violin(). All aesthetics are identical to the box plot above. If you want to includes visualisations of each individual data point, you can use geom_jitter(). You can also visualise the mean and standard error using the stat_summary() function.

ggplot(data1.clean,aes(x = cat.dog,y = introvert,fill = cat.dog)) +
  geom_violin() +
  stat_summary() +
  geom_jitter(width = .1,alpha = .2) +
  theme_classic()

4. Write-up analysis.

In order to report a t-test in APA format, we need the following information:

The t-statistic and degrees of freedom (df) for the statistical test.
the p-value from the statistical test.
The mean on the DV for both groups.
The standard deviations on the DV for both groups.

The first three items can be obtained from the output of t.test() above. However, to get the SDs of the two groups, we need to cover a few more tidyverse functions.

As covered in Workbook 2, summarise() can be used to get descriptive statistics by using functions like mean() and sd() to get the mean and standard deviation of a variable respectively. However, in order to calculate these statistics separately for different groups, we need to use the group_by() function. group_by() will group data based on a column, and perform all subsequent tidyverse functions separately for each group until you use the ungroup() function.

data1.summary <- data1.clean %>%
  group_by(cat.dog) %>%
  summarise(introvert_mean = mean(introvert,na.rm = TRUE),
            introvert_sd = sd(introvert,na.rm = TRUE))

data1.summary

## # A tibble: 2 × 3
##   cat.dog introvert_mean introvert_sd
##   <chr>            <dbl>        <dbl>
## 1 cat               13.4         2.96
## 2 dog               13.1         4.06

Once we have all the required information, we can write-up! A few things to note about APA formatting:

All numerical values are rounded to 2 decimal points, except p-values which are rounded to 3 decimal points.
All letters indicating stats (e.g., M, SD, t, p) should be italicised.

An independent-samples t-test found that there was no significant difference on introversion between cat-people (M = 13.38, SD = 2.96) and dog-people (M = 13.11, SD = 4.06), t(29.25) = 0.26, p = 0.799.

Bar Graphs

A common way of reporting data from a t-test is a bar graph. Unfortunately, this is a very uninformative way of presenting data (it gives no representation for the spread/variability in the data). We provide the code here so we you can re-create a bar graph if needed, and know how to read one if you see one in a paper, but then we will never speak of this evil again. If you need to plot data with a categorical IV and continuous DV, aim to use a box plot, or better yet a violin plot.

In a bar graph, the height of the bars represent the mean for each group. If you’re lucky, you may get error bars, which usually represents the standard error. In the example below, the error bars represents the standard error of the mean for each group on the introversion scale. Note that we have to calculate the standard error ourselves.

data1.summary <- data1.clean %>%
  group_by(cat.dog) %>%
  summarise(introvert_mean = mean(introvert,na.rm = TRUE),
            introvert_sd = sd(introvert,na.rm = TRUE),
            introvert_se = introvert_sd/sqrt(n()))

ggplot(data1.summary,aes(x = cat.dog,y = introvert_mean,fill = cat.dog)) +
  geom_col() +
  geom_errorbar(aes(ymin = introvert_mean - introvert_se,ymax = introvert_mean + introvert_se),width = .2) +
  theme_classic()

One-way Between-Subjects ANOVA (Analysis of Variance)

The one-way ANOVA is used when you have a continuous DV and a categorical IV with more than two groups. Let’s revisit the hypothesis that cat-people are more introverted than dog-people, but what if we did not exclude participants that reported being ‘both’ or ‘neither’? We would then have four groups in the IV; therefore, a one-way ANOVA is the appropriate statistical test.

1. Prepare the data for analysis.

The code below is identical to the code above, except we do not exclude participants who reported “both” or “neither” (these commands have been commented out below).

data3.clean <- data %>%
#  filter(cat.dog != "both") %>%
#  filter(cat.dog != "neither") %>%
   filter(cat.dog != "") %>%
  mutate(introvert = introversion2 + introversion5 + introversion7 + introversion8 + introversion10) %>%
  select(cat.dog,introvert)

2. Run the statistical test.

We run the ANOVA using the aov() function. We can then pipe it to the summary() function to get interpretable output.

aov(introvert ~ cat.dog,data = data3.clean) %>%
  summary()

##             Df Sum Sq Mean Sq F value Pr(>F)
## cat.dog      3    3.8   1.251   0.097  0.962
## Residuals   76  983.4  12.940

The table above tells us whether there is a significant difference between any of the groups, but it does not tell us which groups are significantly different. In order to do this, we need to do follow-up t-tests. You only need to conduct follow up comparisons if the ANOVA is significant. Even though we do not find any significant differences above, we will go through the process anyway from demonstrative purposes.

In the code below, we run a t.test similar to what we did above. However, when calling the data, we use the filter function to only select participants who are in one of two groups. The | symbol means ‘or’, so below, we are only analysing the data of participants who are in the dog ‘or’ cat group.

t.test(introvert ~ cat.dog,data = filter(data3.clean,cat.dog == "dog" | cat.dog == "cat"))

## 
##  Welch Two Sample t-test
## 
## data:  introvert by cat.dog
## t = 0.25707, df = 29.246, p-value = 0.7989
## alternative hypothesis: true difference in means between group cat and group dog is not equal to 0
## 95 percent confidence interval:
##  -1.901682  2.448691
## sample estimates:
## mean in group cat mean in group dog 
##          13.38462          13.11111

Now, we are only including participants in the dog or both group:

t.test(introvert ~ cat.dog,data = filter(data3.clean,cat.dog == "dog" | cat.dog == "both"))

## 
##  Welch Two Sample t-test
## 
## data:  introvert by cat.dog
## t = 0.024921, df = 49.294, p-value = 0.9802
## alternative hypothesis: true difference in means between group both and group dog is not equal to 0
## 95 percent confidence interval:
##  -2.010723  2.061228
## sample estimates:
## mean in group both  mean in group dog 
##           13.13636           13.11111

We continue to repeat this process until we have tested all possible linear combinations:

t.test(introvert ~ cat.dog,data = filter(data3.clean,cat.dog == "dog" | cat.dog == "neither"))

## 
##  Welch Two Sample t-test
## 
## data:  introvert by cat.dog
## t = 0.55857, df = 23.735, p-value = 0.5817
## alternative hypothesis: true difference in means between group dog and group neither is not equal to 0
## 95 percent confidence interval:
##  -1.498424  2.609536
## sample estimates:
##     mean in group dog mean in group neither 
##              13.11111              12.55556

t.test(introvert ~ cat.dog,data = filter(data3.clean,cat.dog == "cat" | cat.dog == "both"))

## 
##  Welch Two Sample t-test
## 
## data:  introvert by cat.dog
## t = -0.22276, df = 28.998, p-value = 0.8253
## alternative hypothesis: true difference in means between group both and group cat is not equal to 0
## 95 percent confidence interval:
##  -2.527520  2.031016
## sample estimates:
## mean in group both  mean in group cat 
##           13.13636           13.38462

t.test(introvert ~ cat.dog,data = filter(data3.clean,cat.dog == "cat" | cat.dog == "neither"))

## 
##  Welch Two Sample t-test
## 
## data:  introvert by cat.dog
## t = 0.75543, df = 19.863, p-value = 0.4589
## alternative hypothesis: true difference in means between group cat and group neither is not equal to 0
## 95 percent confidence interval:
##  -1.461233  3.119353
## sample estimates:
##     mean in group cat mean in group neither 
##              13.38462              12.55556

t.test(introvert ~ cat.dog,data = filter(data3.clean,cat.dog == "both" | cat.dog == "neither"))

## 
##  Welch Two Sample t-test
## 
## data:  introvert by cat.dog
## t = 0.55397, df = 23.875, p-value = 0.5848
## alternative hypothesis: true difference in means between group both and group neither is not equal to 0
## 95 percent confidence interval:
##  -1.583681  2.745297
## sample estimates:
##    mean in group both mean in group neither 
##              13.13636              12.55556

Remember, in your interpretation to adjust the significance level (alpha) to account for family-wise error rate. Since we conducted six comparisons, the significance level should be reduced to 0.008 (that is, .05 divided by 6).

3. Visualise the Data.

Similar to the independent-samples t-test, we can use a violin plot to visualise the distributions for each group.

ggplot(data3.clean,aes(x = cat.dog,y = introvert,fill = cat.dog)) +
  geom_violin() +
  stat_summary() +
  geom_jitter(width = .2,alpha = .2) +
  theme_classic()

4. Write-up analysis.

In order to report a one-way ANOVA in APA format, we need the following information:

The F-statistic for the statistical test.
The Group and Residual degrees-of-freedom.
the p-value from the statistical test.

If the ANOVA is significant, then you need to also include the post-hoc comparisons. For this, you need the following information (essentially the same as if you were writing up a t-test):

The mean on the DV for all groups.
The standard deviations on the DV for both groups.
The test-statistics and p-values from the comparisons.

The code below will calculate the mean and standard deviation on the dependent variable for all groups. It is identical to the code that does this for the t-test.

data3.clean %>%
  group_by(cat.dog) %>%
  summarise(dv_mean = mean(introvert,na.rm = TRUE),
            dv_sd = sd(introvert,na.rm = TRUE))

## # A tibble: 4 × 3
##   cat.dog dv_mean dv_sd
##   <chr>     <dbl> <dbl>
## 1 both       13.1  3.54
## 2 cat        13.4  2.96
## 3 dog        13.1  4.06
## 4 neither    12.6  2.19

Once you have all this information, the write-up becomes:

A one-way between subjects ANOVA found no significant difference between groups on introversion, F(3, 76) = 0.1, p = 0.962.

Paired-samples t-test

As discussed in the lecture, a paired-samples t-test is like an independent-samples t-test, except rather than assess two separate groups of participants (i.e., a between-subjects design), the paired samples t-test is used when data is collected from the same participant in two separate conditions (i.e., a within-subjects design). A common example of this is testing for significant differences in an outcome variable before and after a treatment or intervention.

In the example below, we test whether mood is influenced by exposure to a cute cat video. Click here to see that video.. We would expect that mood would improve after viewing the cat video.

1. Prepare the data for analysis.

When conducting a statistical test, we follow the same procedure as before. First, we must clean the data. Below, we select the columns we need to run the paired-samples t-test and remove any participants with missing data on either variable.

data2.clean <- data %>%
  filter(!is.na(pre.mood)) %>%
  filter(!is.na(post.mood)) %>%
  select(student.no,pre.mood,post.mood)

2. Run the statistical test.

We use the same function as with the independent-samples t-test to run a paired-samples t-test, though the formula looks slightly different. Rather than have the DV on the left, and the IV on the right, we specify both variables on the left within a Pair() function. You could consider this is because both columns technically contains values of the DV (just at different levels of the IV of time). On the right side of the ~ symbol, we have a 1.

t.test(Pair(pre.mood,post.mood) ~ 1,data = data2.clean)

## 
##  Paired t-test
## 
## data:  Pair(pre.mood, post.mood)
## t = -4.7804, df = 79, p-value = 7.957e-06
## alternative hypothesis: true mean difference is not equal to 0
## 95 percent confidence interval:
##  -10.746744  -4.428256
## sample estimates:
## mean difference 
##         -7.5875

Alternatively, if you don’t want to use a formula, you can also use the following technique, but note that the arguments supplied here are different to almost all other analyses we will be covering in this module. The first and second arguments are the columns we want to compare. In this example, this is the ‘before’ and ‘after’ column. How we specify this is using the following syntax: data$pre.mood. This is short-hand for specifying the ‘pre.mood’ column in the ‘data’ data.frame. We also have to set ‘paired’ to TRUE so that the function knows it’s a paired-sample t-test. Note, though, we do not have to supply a data.frame using this method. So the code looks like:

t.test(data$pre.mood,data$post.mood,paired = TRUE)

## 
##  Paired t-test
## 
## data:  data$pre.mood and data$post.mood
## t = -4.7804, df = 79, p-value = 7.957e-06
## alternative hypothesis: true mean difference is not equal to 0
## 95 percent confidence interval:
##  -10.746744  -4.428256
## sample estimates:
## mean difference 
##         -7.5875

3. Visualise the data.

We can plot the data using a violin plot, the same way we visualised the data for an independent-samples t-test. However, this does not give an indication of the change in scores within a participant. One way to represent this graphically, is to plot the data point for each participant, and include a line between their scores across the two conditions. We also adjust the opacity of the points and lines using the alpha aesthetic, just so that the graph isn’t a complete mess.

To create this plot, we first need to re-shape the data using the gather() function so that each observation is on a separate row (rather than each participant). The gather function works by first specifying a column name that distinguishes between observations made by the same participant (i.e., between pre.mood and post.mood), then a variable name that corresponds with the response (i.e., the number corresponding with mood). Then, you specify the variables that you want to reshape. Note: The last line in the code that reshapes the data below is only used so that the pre-mood and post-mood variables are in the right order when we plot the data.

plot.data <- data2.clean %>%
  gather(key = "time",value = "mood",pre.mood,post.mood) %>%
  mutate(time = factor(time,levels = c("pre.mood","post.mood")))

ggplot(data = plot.data,aes(x = time,y = mood)) +
  geom_violin(aes(fill = time)) +
  stat_summary() +
  geom_point(alpha = .2) +
  geom_line(aes(group = student.no),alpha = .2) +
  xlab("Exposure Time (Before or After Video)") +
  ylab("Mood Rating") +
  theme(legend.position = "none") +
  theme_classic()

4. Write-up analysis.

Similar to the independent-samples t-test, we require information on the mean and standard deviation for each condition. This code is exactly the same as above.

data2.summary <- data2.clean %>%
  summarise(pre.mean = mean(pre.mood),
            pre.sd = sd(pre.mood),
            post.mean = mean(pre.mood),
            post.sd = sd(post.mood))

data2.summary

##   pre.mean   pre.sd post.mean  post.sd
## 1  64.3125 18.13005   64.3125 16.77294

Or if you have not reshaped the data:

data2.summary <- data2.clean %>%
  summarise(pre.mean = mean(pre.mood,na.rm = TRUE),
            pre.sd = sd(pre.mood,na.rm = TRUE),
            pst.mean = mean(post.mood,na.rm = TRUE),
            pst.sd = sd(post.mood,na.rm = TRUE))

data2.summary

##   pre.mean   pre.sd pst.mean   pst.sd
## 1  64.3125 18.13005     71.9 16.77294

A paired-samples t-test found that there was a significant difference on mood before (M = 64.31, SD = 18.13) and after (M = 71.9 , SD = 16.77) viewing a cute cat video, t(79) = -4.78, p < .001.

HPSP131 - Workbook 5 - The General Linear Model: t-tests and ANOVA

Objectives

Class Data

Exercises

Content

Before we begin…

The General Linear Model in R

Formulas

Independent-samples t-test

1. Prepare the data for analysis.

2. Run the statistical test.

3. Visualise the data.

Box Plot

Violin Plot

4. Write-up analysis.

Bar Graphs

One-way Between-Subjects ANOVA (Analysis of Variance)

1. Prepare the data for analysis.

2. Run the statistical test.

3. Visualise the Data.

4. Write-up analysis.

Paired-samples t-test

1. Prepare the data for analysis.

2. Run the statistical test.

3. Visualise the data.

4. Write-up analysis.

Exercises