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Open Lecture on Statistics
Statistical notes for clinical researchers: the independent samples
t -test -
Hae-Young Kim
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Restor Dent Endod 2019;44(3):e26.
DOI: https://doi.org/10.5395/rde.2019.44.e26
Published online: July 17, 2019
Department of Health Policy and Management, College of Health Science, andDepartment of Public Health Science, Graduate School, Korea University, Seoul,Korea. JFIF d d Ducky d qhttp://ns.adobe.com/xap/1.0/ |er˭|I|-fGK.rΛp8XV1%K6mvG+tc+qE &ǸC_Nm:l=_/m5^[dߌڇ.c<%:)tQ$Ow~-aY;UJ >=F)2[nk؆?훐 M=l6[4(O.]2#-H^n#->&mp5~Fӛ+|| S,xag% qkEUzUgæBhߕP(7]kFnq?֖Cpru Z6*rEڊtS|*tI*E}7R<,nUU֫^I7Q*mSly%rdȓd8hE<9oHhMfNSRj[i7D[Rj݊+kდq{"$$H?p \̅S?㭻;t~ R߁ )^/>Qj`yt[w ԛ;² ~+ ߔ_ YW~|o] ?x^ᯛ `ʼn;g)T@vWn]>&4l p+$D̢1l|ȨF%-} .9[}w~
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- Correspondence to Hae-Young Kim, DDS, PhD. Professor, Department of Health Policy and Management, Korea University College of Health Science, and Department of Public Health Science, Korea University Graduate School, 145 Anam-ro, Seongbuk-gu, Seoul 02841, Korea. kimhaey@korea.ac.kr
Copyright © 2019. The Korean Academy of Conservative Dentistry
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (https://creativecommons.org/licenses/by-nc/4.0/) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
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The t-test is frequently used in comparing 2 group means. The compared
groups may be independent to each other such as men and women. Otherwise, compared data
are correlated in a case such as comparison of blood pressure levels from the same
person before and after medication (Figure 1). In
this section we will focus on independent t-test only. There are 2
kinds of independent t-test depending on whether 2 group variances can
be assumed equal or not. The t-test is based on the inference using
t-distribution.
T-DISTRIBUTION
The t-distribution was invented in 1908 by William Sealy Gosset, who
was working for the Guinness brewery in Dublin, Ireland. As the Guinness brewery did
not permit their employee's publishing the research results related to their work,
Gosset published his findings by a pseudonym, “Student.” Therefore,
the distribution he suggested was called as Student's
t-distribution. The t-distribution is a
distribution similar to the standard normal distribution,
z-distribution, but has lower peak and higher tail compared to it
(Figure 2).
According to the sampling theory, when samples are drawn from a normal-distributed
population, the distribution of sample means is expected to be a normal
distribution. When we know the variance of population, σ2, we can
define the distribution of sample means as a normal distribution and adopt
z-distribution in statistical inference. However, in reality,
we generally never know σ2, we use sample variance,
s2, instead. Although the
s2 is the best estimator for σ2,
the degree of accuracy of s2 depends on the sample size.
When the sample size is large enough (e.g., n =
300), we expect that the sample variance would be very similar to the population
variance. However, when sample size is small, such as n = 10, we
could guess that the accuracy of sample variance may be not that high. The
t-distribution reflects this difference of uncertainty
according to sample size. Therefore the shape of t-distribution
changes by the degree of freedom (df), which is sample size minus one (n − 1)
when one sample mean is tested.
The t-distribution appears to be a family of distribution of which
shape varies according to its df (Figure 2).
When df is smaller, the t-distribution has lower peak and higher
tail compared to those with higher df. The shape of t-distribution
approaches to z-distribution as df increases. When df gets large
enough, e.g., n = 300,
t-distribution is almost identical with
z-distribution. For the inferences of means using small samples, it
is necessary to apply t-distribution, while similar inference can
be obtain by either t-distribution or
z-distribution for a case with a large sample. For inference of 2
means, we generally use t-test based on
t-distribution regardless of the sizes of sample because it is
always safe, not only for a test with small df but also for that with large df.
INDEPENDENT SAMPLES T-TEST
To adopt z- or t-distribution for inference using
small samples, a basic assumption is that the distribution of population is not
significantly different from normal distribution. As seen in Appendix 1, the normality assumption needs to be tested in
advance. If normality assumption cannot be met and we have a small sample
(n < 25), then we are not permitted to use
‘parametric’ t-test. Instead, a non-parametric
analysis such as Mann-Whitney U test should be selected.
For comparison of 2 independent group means, we can use a
z-statistic to test the hypothesis of equal population means only
if we know the population variances of 2 groups, σ 1 2 
and σ 2 2 
, as follows;
and , as follows;where X̄1 and
X̄2, σ 1 2 σ 2 2
and , and n1 and
n2 are sample means, population variances, and the
sizes of 2 groups.Again, as we never know the population variances, we need to use sample variances as
their estimates. There are 2 methods whether 2 population variances could be assumed
equal or not. Under assumption of equal variances, the t-test
devised by Gosset in 1908, Student's t-test, can be applied. The
other version is Welch's t-test introduced in 1947, for the cases
where the assumption of equal variances cannot be accepted because quite a big
difference is observed between 2 sample variances.
In Student's t-test, the population variances are assumed equal.
Therefore, we need only one common variance estimate for 2 groups. The common
variance estimate is calculated as a pooled variance, a weighted average of 2
sample variances as follows;
where s 1 2 
and s 2 2 
are sample variances.
and are sample variances.The resulting t-test statistic is a form that both the
population variances, σ 1 2 σ 1 2 s p 2 
. The df is given as
n1 + n2 − 2
for the t-test statistic.
and , are exchanged with a common variance
estimate, . The df is given as
n1 + n2 − 2
for the t-test statistic.In Appendix 1, ‘(E-1) Leven's test
for equality of variances’ shows that the null hypothesis of equal
variances was accepted by the high p value, 0.334 (under
heading of Sig.). In ‘(E-2) t-test for equality of means
t-values’, the upper line shows the result of
Student's t-test. The t-value and df are shown
−3.357 and 18. We can get the same figures using the formulas Eq. 2 and
Eq. 3, and descriptive statistics in Table
1, as follows.
The result of calculation is a little different from that by SPSS (IBM Corp.,
Armonk, NY, USA) of Appendix 1, maybe
because of rounding errors.
Actually there are a lot of cases where the equal variance cannot be assumed.
Even if it is unlikely to assume equal variances, we still compare 2 independent
group means by performing the Welch's t-test. Welch's
t-test is more reliable when the 2 samples have unequal
variances and/or unequal sample sizes. We need to maintain the assumption of
normality.
Because the population variances are not equal, we have to estimate them
separately by 2 sample variances, s 1 2 s 2 2
and . As the result, the form of
t-test statistic is given as follows;where ν is Satterthwaite degrees of freedom.
In Appendix 1, ‘(E-1) Leven's test
for equality of variances’ shows an equal variance can be successfully
assumed (p = 0.334). Therefore, the Welch's
t-test is inappropriate for this data. Only for the purpose of
exercise, we can try to interpret the results of Welch's t-test
shown in the lower line in ‘(E-2) t-test for equality of
means t-values’. The t-value and df are
shown as −3.357 and 16.875.
We've confirmed nearly same results by calculation using the formula and by SPSS
software.
The t-test is one of frequently used analysis methods for
comparing 2 group means. However, sometimes we forget the underlying assumptions
such as normality assumption or miss the meaning of equal variance assumption.
Especially when we have a small sample, we need to check normality assumption
first and make a decision between the parametric t-test and the
nonparametric Mann-Whitney U test. Also, we need to assess the
assumption of equal variances and select either Student's
t-test or Welch's t-test.
Appendix
Tables & Figures
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XML DownloadStatistical notes for clinical researchers: the independent samples
t -test
Figure 1 Types of 2-sample t-test.
Figure 2 The t-distribution with various degrees of freedom (df)
compared to z-distribution.
Figure 1
Figure 2
Statistical notes for clinical researchers: the independent samples
t -test
Descriptive statistics and result of the Student's t-test
| Group | No. | Mean | Standard deviation | p value |
|---|---|---|---|---|
| 1 | 10 | 10.28 | 0.5978 | 0.004 |
| 2 | 10 | 11.08 | 0.4590 |
Table 1 Descriptive statistics and result of the Student's
t-test
