7. t-distribution, confidence intervals and t-tests
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 sample mean and 𝑠 𝑥̅ is its associated standard error. Remember here, that 𝑥̅ is
the estimate of population mean and 𝑠 𝑥̅ quantifies its accuracy. As a result, the t-value
represents the estimate of 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 tdistribution.
Its shape is closely similar to Z (standard normal distribution). In contrast to Z, t
distribution has a single parameter – number of degrees of freedom, which equals number
of observations in given sample minus 1. In fact, t approaches Z asymptotically for high DF
(Fig 7.1). Similarly, to normal distribution, 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. t-distribution with its two tails and 2.5% and 97.5%-quantiles.
Confidence intervals for mean value and single sample t-test
t-distribution can be used to compute confidence intervals (CI), i.e. intervals within which
the population mean value lies with 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 population mean is significantly
different 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 single sample
t-test, which tests a null hypothesis that given values X equals the populations mean. Using
the formula for t-value, and DF, the t-test determines type I error probability associated with
rejection of such hypothesis.
Student t-test
If means can be compared with an a-priori given value, two means of different samples
should also be comparable with each other. This is done by 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 standard error of their
difference. This is then computed using following formula:
𝑠 𝑥̅1−𝑥̅2
= √
𝑠 𝑝
2
𝑛1
+
𝑠 𝑝
2
𝑛2
where 𝑠 𝑝
2
is 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. Equality of
population variances, called homogeneity of variance is one of the t-test assumptions. In
addition, t-test assumes that the samples come from populations that are normally
distributed. There is also the universal assumption that individual observations are
independent.
t-test is relatively robust to violations of the assumptions about 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 t-value
computation and also the degrees of freedom are approximated (as a result, DF is usually
not an integer). Note, that Welch t-test is used by default in R. In original (two-sample)
Student t-test, the DF is determined as
DF = n1 – 1 + n2 – 1
Paired t-test
Paired t-test is used to analyzed data composed of paired observations. For instance,
difference of length between left and right arms of people would be analyzed by a paired ttest.
Null hypothesis in this case is that the difference within the pair is zero. In fact, paired ttest
is fully equivalent to single sample t-test comparing the within-pair difference
distribution with zero. Because in paired t-test, there is just one sample (of paired values) DF
= n – 1.
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 lower confidence
limit and the mean.
2. t-test
Function t.test. For two sample, 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 for
single sample test)
paired – TRUE specifies a paired t-test analysis.