• A little simulation of why we divide by \(n-1\), not \(n\) when calculating the standard deviation
• Quick recap of the box1-box2 example
• What is a t-test?
• What different t-tests exist?
• What is a t-distribution?
• What is one-tailed vs. two-tailed testing?
• How to do a t-test in R
• What assumptions need to be met?

pnorm(q = 160, mean = 164, sd = 6)
[1] 0.2524925

Is a value that we’d get 25% of the time by chance rare?

• Only one response variable, one predictor
• The predictor variable is binomial (e.g. treated, non-treated)
• For example, we can ask if the movie ‘Scream 2’ is scarier than the original ‘Scream’.
• We could measure heart rates (which indicate anxiety) during both films and compare them.

This situation can be analysed with a t-test:

scream1 = c(180, 165, 122, 156, 170) #max heart rates scream1
scream2 = c(190, 145, 100, 138, 166) #max heart rates scream2
t.test(scream1, scream2)

• Independent t-test (or simply t-test)
  • Compares two means based on independent data.
  • E.g. data from different groups of people
• Paired (or dependent) t-test
  • Compares two means based on related data.
  • E.g. data from the same people measured at different times.
  • Data from ‘matched’ samples (e.g. before - after)
• One-tailed vs. two-tailed testing

We need a metric (a test statistic!) that puts the difference between the samples into perspective with

• the difference between the samples that we would expect by chance, and
• the standard deviations of the two samples

This is called the t-statistic:

\[t = \frac{\text{observed difference - expected difference}}{\text{estimate of the standard deviations}}\]

In fact, the expected difference is mostly zero (this is the case in the following examples)

\[ t = \frac{\bar{X_1}-\bar{X_2}}{\sqrt{\frac{s^2_p}{n_1} + \frac{s^2_p}{n_2}}} \]

\[ s^2_p = \frac{(n_1 - 1)s^2_1 + (n_2 - 1)s^2_2}{n_1 + n_2 -2} \]

\(t\) follows a \(t\)-distribution!

\(n_1 + n_2 - 2\) are the degrees of freedom, the only parameter for this distribution

Use the equivalent commands to rnorm(), pnorm(), and qnorm()

rt(100, df = 10); pt(q = 0, df = 10); qt(p = .025, df = 10)

You can now organise your data in two ways, the so-called ‘wide’- or ‘long’ format:

d_wide = data.frame(realspider, spiderpicture)
d_long = data.frame(treat = c(0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1),
                     anxiety = c(realspider, spiderpicture))
head(d_wide, 3)
  realspider spiderpicture
1          3             5
2          5             6
3          3             3
head(d_long, 3)
  treat anxiety
1     0       3
2     0       5
3     0       3

• To do a t-test we use the function t.test()
• Depending on the format of your data, you can use this function in two ways:

t.test(d_wide$realspider, d_wide$spiderpicture)
t.test(d_long$anxiety ~ d_long$treat) #or:
t.test(anxiety ~ treat, data = d_long) #preferred

What does the dollar sign do again…?

I recommend the long format, it is more versatile and easier to use when you have a lot of data!

The result however looks the same:

t.test(d_long$anxiety ~ d_long$treat)

    Welch Two Sample t-test

data:  d_long$anxiety by d_long$treat
t = -0.86966, df = 9.9746, p-value = 0.4049
alternative hypothesis: true difference in means between group 0 and group 1 is not equal to 0
95 percent confidence interval:
 -3.562974  1.562974
sample estimates:
mean in group 0 mean in group 1 
       5.166667        6.166667 

How do we interpret this output?

• Apart from a recall of the data, the group means and a confidence interval for the difference between groups, we obtain a t-value, the degrees of freedom, and a p-value

• We now compare our obtained t-value against a random t-distribution: how rare is our t-value, which reflects the difference between the groups and their standard deviations?

pt(q = -0.87, df = 10)
[1] 0.20235

This is our p-value! (in fact we need to multiply it by 2 to account for the two tails)

qt(p = .025, df = 10)
[1] -2.228139

We fail to reject our null hypotheis, but we cannot accept our null hypothesis

We cannot state that there is no difference in anxiety between seeing a real spider or a picture of a spider

We can only say that we don’t have enough evidence to reject the null hypothesis, as we could be committing a type II error, and we don’t know the probability for such an error (\(\beta\), see slides on power test!)

On average, participants did not experience greater anxiety from real spiders (6.1 \(\pm\) 0.79 s.e.) than from pictures of spiders (5.1 \(\pm\) 0.83 s.e.; t-test, p = 0.4).

or, if it were significant:

On average, participants did experience greater anxiety from real spiders (xy \(\pm\) xy s.e.) than from pictures of spiders (xy \(\pm\) xy s.e.; t-test, p = xyz).

• Does plant transpiration respond to treatment with carbon dioxide?
• Null hypothesis: plant transpiration is not affected by carbon dioxide
• Measure transpiration in 12 leaves before and after treatment with carbon dioxide
• The measurements are paired: the 12 leaves are the same before and after the treatment
• Could you create a fictitious data frame for this example?

d1 = data.frame(transpiration = c(2, 4, 3, 4, 3, 5, 5, 4, 3, 6, 5, 4, 1, 2, 1, 4, 2, 3, 4, 3, 3, 2, 1, 4),
                co2 = rep(c('before', 'after'), each = 12))
head(d1)
  transpiration    co2
1             2 before
2             4 before
3             3 before
4             4 before
5             3 before
6             5 before

t.test(d1$transpiration ~ d1$co2, paired = T)

    Paired t-test

data:  d1$transpiration by d1$co2
t = -3.7607, df = 11, p-value = 0.003151
alternative hypothesis: true mean difference is not equal to 0
95 percent confidence interval:
 -2.3778894 -0.6221106
sample estimates:
mean difference 
           -1.5 
#set argument paired to TRUE

• Depending on whether we expect differences between groups to occur in both directions or only in one direction, we use 1- or 2-tailed t-tests
• In the spider example, differences could occur in both directions (those shown a picture could be more afraid): 2-tailed
• If you test whether carrying backpacks makes people shorter (paired, before and after): 1-tailed (carrying backpacks can’t make you taller!

In R, choose alternative = 'two-sided', 'greater' or 'less' (by default the argument is set to ‘two-sided’:

t.test(d1$transpiration ~ d1$co2, paired = T, alternative = 'greater')

• Both the independent t-test and the paired t-test are parametric tests based on the normal distribution. Therefore, they assume:
  • The sampling distribution is normally distributed. In the paired t-test this means that the sampling distribution of the differences between scores should be normal, not the scores themselves (assumption of normality).
  • Variances in the two samples are roughly equal (assumption of homogeneity of variance). This assumption however can be relaxed if variance heterogeneity is accounted for, and by default, R will do that!
  • Scores are independent (unless in a paired t-test) (assumption of independence)

• How to compute a t-test
• What an independent and a paired t-test is, how they are used
  • how to formulate a null hypothesis for a t-test
  • how to run a t-test in R
  • how to interpret the output of a t-test
• How the argument ‘alternative’ is used (one- and two-tailed tests)
• What the t-distribution is, how it relates to the normal distribution
• What it means to commit a type I or type II error in a t-test
• What assumptions need to be met in a t-test

• t-statistic
• t-distribution
• t-test
• independent t-test
• paired t-test
• one-tailed t-test
• two-tailed t-test
• p-value