7. The t-distribution, confidence intervals, and t-tests
The t-distribution
For any fixed value X, a t-value can be computed from a sample of a quantitative random
variable using this formula:
𝑡 =
𝑋 − 𝑥̅
𝑠 𝑥̅
where, 𝑥̅ is the sample mean and 𝑠 𝑥̅ is its associated standard error. Recall here that 𝑥̅ is the
estimate of the population mean and 𝑠 𝑥̅ quantifies its accuracy. As a result, the t-value
represents the estimate of the difference between X and the population mean. Because 𝑥̅
is a random variable, t-value is also a random variable, and its probability distribution is
called the t-distribution. Its shape is closely similar to Z (standard normal distribution). In
contrast to Z, the t distribution has a single parameter – the number of degrees of freedom,
which equals the number of observations in a given sample minus 1. In fact, t approaches Z
asymptotically for high df (Fig 7.1). Similarly to the normal distribution, the t-distribution is
symmetric, and its two tails must be considered when computing probabilities {Fig 7.2).
Fig. 7.1 Probability density plot of t-distributions with different DF and their comparison to
standard normal distribution (Z).
Fig 7.2. The t-distribution with its two tails and the 2.5% and 97.5%-quantiles.
Confidence intervals for the mean value and single sample t-test
The t-distribution can be used to compute confidence intervals (CI), i.e. intervals within
which the population mean value lies with a certain probability (usually 95%). The
confidence limits (CL) within which the CI lies are determined using these formulae:
𝐶𝐿𝑙𝑜𝑤 = 𝑥̅ + 𝑡(𝑑𝑓,𝑝=0.025) 𝑠 𝑥̅
𝐶𝐿ℎ𝑖𝑔ℎ = 𝑥̅ + 𝑡(𝑑𝑓,𝑝=0.975) 𝑠 𝑥̅
where t(df, p) equals 2.5% or 97.5% probability quantile of t-distribution with given df. These
intervals can be used as error bars in barplots or dotcharts. In fact, they represent the best
option to be used like this (in contrast to standard error or 2 x standard error).
Confidence intervals can also be used to determine whether the population mean differs
significantly from a given value: a value lying outside the CI is significantly different (at 5%level
of significance) while a value lying inside is not. This is closely associated with the single
sample t-test, which tests a null hypothesis that a value X equals the population mean. Using
the formula for t-value and DF, the t-test determines the probability of type I error
associated with rejection of such a hypothesis.
Student t-test
If means can be compared with an apriori given value, two means of different samples
should also be comparable. This is done by a two-sample t-test1, which quantifies
uncertainty about the values of both means considered:
𝑡 =
𝑥̅1 − 𝑥̅2
𝑠 𝑥̅1−𝑥̅2
where 𝑥̅1 and 𝑥̅2 are arithmetic means of the two sample and 𝑠 𝑥̅1−𝑥̅2
is the standard error of
their difference. The 𝑠 𝑥̅1−𝑥̅2
is computed using the following formula:
𝑠 𝑥̅1−𝑥̅2
= √
𝑠 𝑝
2
𝑛1
+
𝑠 𝑝
2
𝑛2
where 𝑠 𝑝
2
is the pooled variance of the two samples, and n1 and n2 are sample sizes of the
two samples. Pooling variance like this is only possible if the two variances are equal.
Correspondingly, the equality of population variances, called homogeneity of variance, is
one of the t-test assumptions. In addition, the t-test assumes that the samples come from
populations that are distributed normally. There is also the universal assumption that
individual observations are independent.
The t-test is relatively robust to violations of the assumptions about the homogeneity of
variance and normality (i.e. their moderate violation does not produce strongly biased test
outcomes). If variances are not equal, Welch approximation of t-test (Welch t-test) can be
used instead of the original Student t-test. A slightly modified formula is used for the t-value
computation, and also the degrees of freedom are approximated (as a result, df is usually
not an integer). Note here that the Welch t-test is used by default in R. In the original (twosample)
Student t-test, the DF is determined as
DF = n1 – 1 + n2 – 1
where n1 is the size of sample 1 and n2 is the size of sample 2.
Paired t-test
Paired t-test is used to analysis of data composed of paired observations. For instance, a
difference in length between left and right arms of people would be analyzed using a paired
t-test. The null hypothesis, in this case, is that the difference within the pair is zero. In fact, a
paired t-test is fully equivalent to a single sample t-test comparing the within-pair difference
distribution with zero. The number of degrees of freedom is determined as DF = n – 1
because there is just one sample (of paired values) in a paired t-test.
1
Called also Student t-test after its inventor William Sealy Gosset (1976-1937) who used the pen name
Student.
How to do in R
1. t distribution computations
functions pt and qt are available. For instance, qt(0.025, df)
can be used to compute the difference between the lower
confidence limit and the mean.
2. t-test
Function t.test. For two samples, the best way is to use a
classifying factor and response variable in two columns. Then,
t.test(response~factor) can be used. But t.test(sample1,
sample2) is also okay.
important parameters:
var.equal – switches between Welch and Student variants.
Defaults to FALSE (Welch)
mu – a priori null value of the difference (relevant mainly
for a single sample test)
paired – TRUE specifies a paired t-test analysis.