How to find which group performed better if the null hypothesis is rejected in a t-test?

Question

I have done t-test (two-sample assuming unequal variances). In this case, I am rejecting the null hypothesis. There is a difference between the groups. I am just wondering if there is any way to find which group is performing better?

Answer 1

It depends on how you've set the test up.

If your test statistic has $\bar{x}_1 - \bar{x}_2$ in the numerator, then your test statistic is looking at the mean of group 1 minus the mean of group 2. A negative difference means group 2 has a higher mean.

So to know which group did better, look at the sign of the test statistic and the order in which the group means were used.

Answer 2

The group that has the mean that is in the "better" direction did better on average. The t-test lets you assess the strength of evidence against the null hypothesis, and so if you got a small enough p-value then it is reasonable (in some sort of proportion to the smallness of the p-value) to think that one group did better.

Seeing which group did better on average needs no statistics beyond the mean. Just graph the data. Never make inferences on the basis of data without examining the data in some sort of visual display.

Answer 3

It helps to check the means and standard deviations of your groups as well as to visualize the differences. I really like the Datanovia approach to things, where you also just add test data onto plots. An example is given below using R. First we can load the required libraries

#### Load Libraries ####
library(datarium)
library(tidyverse)
library(rstatix)

#### Check Mean/SD of Groups ####
genderweight %>% 
  group_by(group) %>% 
  summarise(mean.weight = mean(weight),
            sd.weight = sd(weight))

giving us these summary statistics

# A tibble: 2 × 3
  group mean.weight sd.weight
  <fct>       <dbl>     <dbl>
1 F            63.5      2.03
2 M            85.8      4.35

We can see from the descriptive data here that the female group has a lower weight on average and varies less than the male group, so if we have a significant t-test, we should expect that it is because the female group is lower on average than the male group. We can run a t-test to check quickly:

#### Run T-Test ####
t <- genderweight %>% 
  t_test(formula = weight ~ group) 

And indeed this is true...

# A tibble: 1 × 8
  .y.    group1 group2    n1    n2 statistic    df        p
* <chr>  <chr>  <chr>  <int> <int>     <dbl> <dbl>    <dbl>
1 weight F      M         20    20     -20.8  26.9 4.30e-18

The negative sign indicates that the female group (the reference group here) has a lower mean than the male group. We can look at this difference directly with a boxplot and even add our t-test data to contextualize these differences:

#### Visualize Differences ####
genderweight %>% 
  ggplot(aes(x=group,
             y=weight,
             fill=group))+
  geom_boxplot()+
  labs(title = "Weight By Gender",
       subtitle = get_test_label(t, detailed = T))

And now you can see the which group has a smaller and which a larger mean and that the female group has a much narrower box because of its lack of variation compared with males:

Edit

Nick brings up a very compelling point about boxplots in the comments. While this gives some perspective on what the average weights are with each group, they are more based around the interquartile range and medians of the data, which are not directly comparable to means and standard deviations (especially in the case of skewed data). Some other visualizations may serve this purpose better (of which he mentions quantile plots with overlayed means), but at the minimum, inspecting the means and SDs as well as giving yourself graphical representations of the data help.