### t tests

```The t-test
Dr David Field
Summary
• Andy Field, 3rd Edition, chapter 9 covers today's lecture
– But one sample t and formulas are not covered
• The t test for independent samples
– Comparing the means of two samples of subjects
• Degrees of freedom
• The paired samples t test (also known as repeated
measures and related samples)
– Comparing the means of two experimental conditions when the
same sample participated in both conditions
• The one sample t test
– Comparing the mean of a single sample against a fixed value, e.g.
50% chance performance, or 0
• Effect size
– The degree to which changes in a DV are caused by the IV relative
to other sources of variation
• 95% confidence interval of the difference between sample
means
The story so far….
• In workshop 3 you produced an error bar chart for the
blueberries versus sugar cubes with exam scores
experiment
• The 95% confidence intervals around the two means
overlapped, so we could not be sure that the two samples
were from different populations
– We could not reject the null hypothesis
– We could not be sure that we have failed to reject the null
hypothesis
• Today, we will use the t test to make a direct test of the null
hypothesis for that experiment
• By performing a t test you arrive at a significance level or p
value, which is the probability of obtaining the difference
between the means of experimental conditions by random
sampling from a null distribution
The t statistic
• There are several variations on how t is calculated that
depend on the type of experiment (independent samples,
repeated measures, one sample)
• But these are variations on a general theme
• Higher values of t reflect greater statistical significance
Difference between two means
Variation in data
• In the following examples the difference between the means is the
same in all three cases
• But the variability of the samples is different
t will be
smaller than
‘medium’
t will be larger
than ‘medium’
The t statistic
• The size of t can be increased in three ways
– increase the difference between the two sample means
– reduce the variability in the two samples
– Increase the number of participants
Difference between two means
Variation in data
The t statistic
• There are several variations on how t is calculated that
depend on the type of experiment (independent samples,
repeated measures, one sample)
• But these are variations on a general theme
mean sample 1 – mean sample 2
SE of the difference
In different types of experiment different methods are used to calculate
the SE of the difference
SE of the difference (see lecture 4)
• The SE of the difference is similar to the SE of a sample
mean
• It reflects an underlying sampling distribution of many
possible repetitions of the experiment using that exact
sample size, where in each case the difference obtained
between the means would be slightly different
• The key thing is that the SE gets smaller as the sample
size increases and as the underlying SD of the population
of interest reduces
• Because the t formula involves dividing by the SE of the
difference t is in units of SE
– just as Z scores are in units of SD
mean sample 1 – mean sample 2
SE of the difference
SE of the difference for independent samples
• Independent samples (or between subjects) is
where separate samples of participants are tested
in each experimental condition
– We will use the effects of eating blueberries versus
sugar cubes on exam scores as an illustration
• In this case the SE of the difference has to take
into account the fact that there are two samples
contributing to the part of the unsystematic
variation in the scores that is not due to the IV
– IQ and hours studied are things the researchers are not
interested in that might contribute to the variability of
scores in both samples of the blueberry experiment
SE of the difference for independent samples
Sample 1 variance
N sample 1
Sample 2 variance
N sample 2
• N is the number of participants in the groups, which may
be unequal
• Recall that the variance is the sum of squared differences
from the mean divided by N-1 (see lecture 1)
• This formula is very similar to that used in the z test
(lecture 4).
– But the way that t is converted to a p value is slightly different from
the way z is converted to a p value
t formula for independent samples
Sample 1 mean
Sample 2 mean
Sample 1 variance
Sample 2 variance
t=
N sample 1
N sample 2
Blueberry experiment example
t=
50.77
57.47
94.8
30
106.8
30
t = 2.58
Converting t to a p value (significance level)
• The t statistic is a measure of the difference
between the sample means in units of SE (i.e.
units of estimated measurement error)
• It is a measure of how big the effect of the IV is
compared to unsystematic variation
– random allocation of participants to conditions helps to
insure that other sources of variation do not co-vary
with the IV
• Two sample means will differ even under the null
hypothesis, where the only source of variation in
the data is unsystematic
– what is the probability of a value of t this big or bigger
arising under the null hypothesis?
What differs from the z test?
• For the z test, the obtained value of z is referred to the
standard normal distribution, and the proportion of the area
under the curve with that value of z or greater is the p
value
• However, when sample sizes are small the standard
normal distribution is actually a bad model of the null
distribution
– it’s tails are too thin and values close to the mean are too common
• The t distribution has much fatter tails, accurately
representing the chances of large differences between
sample means occurring by chance when samples are
small
– Unlike the standard normal distribution, the t distribution changes
shape as the sample size increases
– As N increases the tails get thinner, and if N > 30, it is almost
identical to the z distribution
The standard normal distribution compared to
the t distribution when N = 2
Blue is the standard
normal distribution
Red is the t probability
distribution if N = 2 in
both groups
The proportion of area
under the curve that
is, e.g., > 2 is much
greater for the t
distribution
The proportion is
14.7% for t and 2.3%
for z
The standard normal distribution compared to
the t distribution when N = 4
Blue is the standard
normal distribution
Red is the t probability
distribution if N = 4
both groups
The proportion of area
under the curve that
is, e.g., > 2 is now
reduced for the t
distribution
The proportion is
5.0% for t and 2.3%
for z
Degrees of Freedom (DOF)
• The shape of the t distribution under the null hypothesis
changes as sample size (N) goes up
• This is because the number of degrees of freedom
increases as the two sample sizes go up
– and the shape of the t distribution is directly specified by the DOF,
not N
• The degrees of freedom for the independent samples t test
are given by
(group one sample size - 1) + (group two sample size -1)
• All statistics have degrees of freedom, and for some
statistics the formula for working them out is more
complicated
• Fortunately, SPSS calculates DOF for you
Degrees of freedom (DOF)
• The DOF are the number of elements in a calculation that
are free to vary.
• If you know only that four numbers have a mean of 10 you
can say nothing about what the four numbers might be
• But if you also know that the first three numbers are 8, 9,
and 11
• Then the fourth number must be 12
• If you only know that the first two numbers are 8 and 9
there are still an infinite possible number that the third and
fourth numbers can take
• Therefore, this calculation of the mean of 4 numbers has 3
DOF
Converting t to a p value (significance level)
• Having worked out the value of t and the number of DOF
associated with the calculation of t, SPSS will report the
exact probability (p value) of obtaining that value of t under
the null hypothesis that both samples were selected
randomly from populations with identical means
• The p value is equal to the proportion of the area under the
curve of the appropriate t distribution where the values are
as big or bigger than the obtained value of t
• In the past you had to look up the p value yourself in
statistical tables of DOF versus t at the back of a
textbook…..
• Remember that if p < 0.05 you can reject the null
hypothesis and say that the effect of the IV on the DV was
statistically significant
Blueberry example
• In the blueberry experiment there was a difference of 6.7%
in the predicted direction, but the 95% confidence intervals
around the means of the two groups overlapped, so it is
useful to perform the t test to see if there is enough
evidence to reject the null hypothesis
• “The effect of supplementing diet with blueberries
compared to sugar cubes on exam scores was statistically
significant, t(58) = 2.58, p = 0.006 (one tailed).”
– This is an incomplete example of reporting experimental results
– A full example will be given later
• We write (one tailed) and divide the p value SPSS gives by
2 because the researchers predicted that blueberries
would improve exam scores, rather than just changing
them
• t is sometimes negative, but this has no special meaning, it
just depends which way round you enter the variables in
the t test dialogue box in SPSS
Related t-test
• This is also known as paired samples or repeated
measures
• Use it when the same sample of participants perform in
both the experimental and control conditions
• For example, if you are interested in whether reaction
times differ in the morning and the afternoon, it would be
more efficient to use the same sample for both
measurements
• If you suspect that practice (or fatigue) effects will occur
when the DV is measured twice in the same person then
counterbalance the order of testing to prevent order effects
from being confounded with the IV
– In the example you’d test half the participants in the morning first
and then in the afternoon, and the other half would be tested in the
afternoon first and then in the morning
– Order effects will then contribute only to unsystematic variation
Advantages of related t over independent t
• In the blueberry experiment, there was variation in scores in
each group due to IQ and hours studied.
– this variation was included in the SE part of the equation
• If you tested the same participants twice, once in the blueberry
condition, and once in the sugar cube condition, employing two
equally difficult exams, then IQ scores and study habits would
be equalised across the two experimental conditions
– This would reduce the risk of a type 1 error occurring due to chance
(sampling) differences in unsystematic variation between conditions
– It would no longer be possible, by chance, for the majority of the higher
IQ and hard working participants to end up in one of the two
experimental conditions
• If you constructed the experiment in this repeated measures
way, and used the independent samples t formula this would
already be an improvement on the separate samples design
• But, a bigger improvement in the sensitivity of the experiment
can be achieved by changing how the SE is calculated
Advantages of related t over independent t
• In the repeated measures design, each score can be
directly paired (compared) with a score in the other
experimental condition
– We can see that John scored 48 in the exam when supplemented
with sugar cubes and 52 when he ate blueberries
– There was no meaningful way to compare the score of an
individual participant with any other individual in the other condition
when independent samples were used
– In the repeated measures case we can calculate a difference score
for John, which is 4
– If Jane scored 70 (sugar cubes) and 72 (blueberries) then her
difference score is 2
– Jane has a very different overall level of performance compared to
John, but we can ignore this, and calculate the error term using
only the difference scores
– In the independent samples t the massive difference between John
and Jane had to be included in the error term
Data table for repeated measures t test
Reaction time Difference (sec)
Participant Reaction
time morning afternoon
[morning –
(sec)
(sec)
afternoon]
Peter
0.90
0.80
-0.10
Sarah
0.40
0.30
-0.10
John
0.30
0.25
-0.05
Jane
0.50
0.45
-0.05
Henry
1.20
1.00
-0.20
MEAN
0.66
0.56
-0.10
Data table for independent samples t test
Participant
(group 1 morning)
Reaction
Participant
time morning (group 2 –
(sec)
afternoon)
Peter
0.90
Tom
Sarah
0.40
Rachel
John
0.30
David
Jane
0.50
Louise
Henry
1.20
James
MEAN
0.66
Reaction time
afternoon (sec)
x
x
x
x
x
1.0
0.80
0.30
0.45
0.25
0.56
t formula for related samples
t=
Mean difference between conditions
SD of the difference
Sample size
(Compare independent samples t)
Sample 1 mean
Sample 2 mean
Sample 1 variance
Sample 2 variance
t=
N sample 1
N sample 2
Worked example of related t
Participant Difference
(sec) [morning
– afternoon]
Peter
-0.10
Sarah
-0.10
John
-0.05
Jane
-0.05
Henry
-0.20
MEAN
-0.10
• The SD of the difference
scores is 0.061 sec and
N is 5
• Therefore, SE is 0.061 /
square root of 5
• 0.061 / 2.23 = 0.027
• t = Mean dif / SE
• -0.10 / 0.027
• t = -3.65
Converting related t to a p value
• The null hypothesis is that the mean difference between
the two conditions in the population is zero
• DOF are given by sample size – 1 for related t
– DOF are needed to determine the exact shape of the probability
distribution, in exactly the same way as for independent samples t
• The value of t is referred to the t distribution, and the p
value is the proportion of the area under the curve equal to
or greater than the obtained t
• This is the probability of a mean difference that big
occurring by chance in a sample if the difference in the
population was zero
• The repeated t test is reported in the same way as the
independent t test
Reaction time example
• “It took longer to respond in the morning than in the
afternoon, t(4) = -3.65, p = 0.02 (two tailed).”
– This is an incomplete example of reporting experimental results
– A full example will be given later
• This is a two tailed test because the researchers thought
that circadian rhythms would result in a different rtm in the
afternoon, but they were not able to predict the direction of
• For a two tailed test you have to consider both tails of the
probability distribution, in this case the area under the
curve that is greater than 3.65 and less than -3.65
• SPSS does this by default
One sample t-test
• Sometimes, you want to test the null hypothesis that the
mean of a single sample is equal to a specific value
• For example, we might hypothesize that, on average,
students drink more alcohol than the people of similar age in
the general population
– Imagine that a previous study used a large and appropriately selected
sample to establish that the mean alcohol consumption per week for
UK adults between the ages of 18 and 22 is 15 units
– We can make use of their large and expensive sample, which
provides a very good estimate of the population mean
– To test the hypothesis we could randomly select a single sample of
students aged 18-22 and compare their alcohol consumption with that
from the published study
• The null hypothesis will be that the mean units of alcohol
consumed per week in the student sample is 15
Data table for one sample t test
Participant Units of
alcohol per
week
Null
hypothesis
Difference (units of
alcohol)
Peter
20
15
5
Sarah
16
15
1
John
10
15
-5
Jane
12
15
-3
Henry
18
15
3
MEAN
15.2
0.2
One sample t formula
t=
Mean difference from null hypothesis
SD of the sample
Sample size
This is the
SE of the
sample
mean
(Compare related samples t formula)
t=
Mean difference between conditions
SD of the difference
Sample size
Worked example of one sample t
Participant Difference
(units of
alcohol)
Peter
5
Sarah
1
John
-5
Jane
-3
Henry
3
MEAN
0.2
• The SD of the difference
scores is 4.14 units and
N is 5
• Therefore, SE is 4.41 /
square root of 5
• 4.41 / 2.23 = 1.85
• t = Mean dif / SE
• 0.2 / 1.85
• t = 0.107
Converting one sample t to a p value
• The null hypothesis is that the population mean is
the same as the fixed test value (15 in this case)
• DOF are given by sample size – 1
– DOF are needed to determine the exact shape of the
probability distribution, in exactly the same way as for
independent samples t
• The value of t is referred to the t distribution, and
the p value is the proportion of the area under the
curve equal to or greater than the obtained t
• This is the probability of the sample mean being
that different from the fixed test value if the true
population mean is equal to the fixed test value
Alcohol consumption example
• “We found no evidence that students consume
more alcohol than other people in their age group,
t(4) = 0.10, p > 0.05, NS”
– This is an incomplete example of reporting experimental
results
– A full example will be given later
• NS is short for for non-significant
• The results of a t test are non significant if p is
larger than 0.05 (5%).
Effect size (Cohen’s d )
• Statistical significance (i.e. p of data under null hypothesis
of < 0.05) is not equivalent to scientific importance
• If SPSS gives a p value of 0.003 for one t test, and 0.03 for
another t test, then the former is “statistically highly
significant” but it is NOT a “bigger effect” or “more
important" than the latter
• This is because the SE of the difference depends partly on
sample size and partly on the SD of the underlying
population (as well as on the size of the mean difference)
– If the sample size is large a trivially small difference between the
means can be statistically highly significant
– If the two underlying populations of scores in an independent
samples t test have a small SD then a trivially small difference in
means will again be statistically highly significant
Effect size (Cohen’s d )
• For these reasons it is becoming increasingly common to
report effect size alongside a null hypothesis test
d=
Condition 1 mean – Condition 2 mean
SD Condition 1 + SD condition 2
2
• The key difference from the t formula is that sample size is
not part of the formula for d
• like z scores, d is expressed in units of SD
• Effect size is basically a z score of the difference
• SPSS does not report effect size as part of the t test
output, but you can easily calculate it yourself
Effect size in the blueberry example
d=
blue 57.47% – sugar 50.77%
Blue SD 10.3% + sugar SD 9.7%
2
• d is 6.7 / 10.05 = 0.66
• This means the mean difference of 6.7% between
the two groups is equal to 0.66 pooled SD
Interpreting effect size
• Cohen (1988)
–
–
–
–
d 0.2 is a small effect
d 0.5 is a medium effect
d 0.8 is a large effect
d of > approx 1.5 is a very large effect
• Warning: Andy Field 3rd Edition uses a different
measure of effect size, called r
• For this course, you must use Cohen’s d
95% confidence interval of the difference
• Previously, you calculated a 95% confidence interval
around a sample mean, and it is also possible to calculate
a 95% confidence interval of the difference between two
sample means
• In the blueberry example, the difference in exam scores
between the two groups was 6.7%
• This difference is actually a point estimate produced by a
single experiment, and in principle there is an underlying
sampling distribution of similar experiments with the same
sample size
– Each time you repeated the experiment the result (mean
difference) would be slightly different
• The point estimate of the difference can be converted to an
interval estimate using a similar method to that for
calculation of an interval estimate based on a sample
mean
95% confidence interval of the difference
• For the blueberry example, SPSS reports a 95%
confidence interval of 1.5 to 11.9 for the underlying mean
difference in the population
• This means that if we replicated the experiment with the
same sample sizes in the sugar and blueberry groups we
could be 95% confident of observing a difference in exam
scores of somewhere between 1.5 and 11.9 percentage
points
• Reporting the 95% confidence interval of the difference is
becoming common practice in journal articles
Reporting t test results
• Here is an example of how to put all the different statistics from
today’s lecture into a paragraph
– “Participants in the blueberry condition achieved higher exam scores
(mean = 57.47%, SD = 10.3%) than those in the sugar cube condition
(mean = 50.77%, SD = 9.7%). The mean difference between conditions
was 6.7%, which is a medium effect size (d = 0.66); the 95%
confidence interval for the estimated population mean difference is
between 1.5 and 11.9%. An independent t test revealed that, if the null
hypothesis were true, such a result would be highly unlikely to have
arisen (t(58) = 2.58; p = 0.006 (one tailed)).”
• This paragraph gives the reader more useful information than the
traditional practice of reporting only the null hypothesis test
• Always report descriptive statistics (mean, SD, effect size,
confidence interval of difference) before inferential statistics that test
the hypothesis and have a p value (for example, t).
Result of facial feedback workshop (Tues)
– “The 38 participants who held the pencil between their
teeth rated the cartoon as funnier (mean = 5.6, SD = 1.80)
than the 42 participants who held the pencil between their
lips (mean = 4.7, SD = 1.78). The mean difference between
conditions was 0.89, which is a medium effect size (d =
0.49); the 95% confidence interval for the estimated
population mean difference is between 0.1 and 1.7. An
independent t test revealed that, if how the pencil was held
had no influence on how the cartoon was perceived, such a
result would be unlikely to have arisen (t(78) = 2.21; p =
0.015 (one tailed)).”
– Note: DOF are (38-1) + (42-1) = 78
– On Monday the group sizes were very unequal (??!!) and
the result was quite different, and non-significant.
```