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218 10 Foreground Toolbox for Hypothesis Testing
The p-value of a test statistic serves as an indicator of the extremity of that statistic
given the sample data. It quantifies the probability of obtaining a more extreme value
than the one observed in the experiment.
The p-value describes how likely we had observed a particular set of data if the
null hypothesis were true. It also tells us, if the null hypothesis of the test was true,
how often we would expect to see a test statistic as extreme or more extreme than
the one calculated by the statistical test. The p-value can vary between 0 and 1.
A p-value lower than significant level (α) is often considered to be statistically
significant. The smaller the p-value, the more unlikely the observed sample and is
stronger evidence more likely to reject the null hypothesis. The greater the p-value
(greater than alpha level), the more likely the observed sample, and there is no
evidence to reject the null hypothesis.
Interpretation of p-value
Statistical significance is a method used to determine whether the p-value of a
statistical test is sufficiently low to warrant rejecting the null hypothesis. But what
constitutes a sufficiently low p-value? This is determined by the alpha level, which
serves as the threshold criterion.
To assess the statistical significance of an observed outcome, it is essential to
compare the p-value with the alpha level.
Tips
Never say “we accept the null hypothesis” instead of “fail to reject the null
hypothesis”, because the decision is based on sample size of n. If the sample
size changes, the conclusion will be different.
The goal of significance tests is to provide evidence to reject the null
hypothesis to demonstrate that it is not a defensible idea, not to accept or
prove it.
Always we are allowed to say:
There is no evidence to reject H
0
or
We are not able to reject H
0
or
We fail to reject
H
0
.
There are two situations that emerge:
If the p-value is very small, less than or equal to pre-defined threshold value (α),
the null hypothesis (
H
0
) is rejected. When this happens, we say that the result is
statistically significant. In other words, we are reasonably sure that there is some
evidence besides chance alone that gave us an observed sample.
If the p-value is greater than pre-defined threshold value alpha (α), we fail to reject
the null hypothesis (don’t say we accept or prove the null hypothesis). In this case,
we say that the result is not statistically significant. In other words, we are reasonably
sure that our observed data can be explained by chance alone.
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10.2 Definitions 219
Tip
Never use general terms such as: non-significant, significant, and highly
significant in your research report or article. Specify the exact p-value on
your report instead of its limits. For example, don’t report p <0.05or p >0.05
instead p = 0.08 or p = 0.15. Because statistical software calculates its exact
value and the reader can better interpret it by seeing the exact value of p.
Basically, statistical significance tests are designed to refute hypotheses. They are
never presented to prove anything. The aim is therefore to demonstrate that the idea
is not defensible. In other words, the tests are designed to reject the issues raised.
Although this point is important both in medical studies and in statistical judgments,
it is unfortunately rarely observed.
Test statistic
A test statistic (without “s” at the end) is a random variable computed from sample
data and serves as a tool for making quantitative decisions regarding hypothesis
testing. Its purpose is to determine whether there is sufficient evidence to “reject” a
hypothesis (null hypothesis) about a process. The test statistic compares the observed
data with what is expected under the null hypothesis and is instrumental in calculating
the p-value. Essentially, a test statistic quantifies the degree of agreement between
a sample of data and the null hypothesis. Its observed value varies randomly across
different random samples and encapsulates information critical for deciding whether
to reject the null hypothesis.
Test statistic is defined as below:
test statistic =
Observed value − Hypothesized value
Observed standard error
,
where the “hypothesized value” is the same value as claimed in the null hypothesis,
and in many cases, it is considered zero.
If the above value is negative, its absolute value should be considered according
to the symmetry property of the normal distribution.
Critical value
In hypothesis testing, a critical value represents a point on the scale of the test statistic
(on the test distribution) that is compared to the observed test statistic to determine
whether to reject the null hypothesis. This critical value serves as a threshold in the
hypothesis test and is contingent upon the significance level (α). It delineates the
boundary of the rejection region and establishes a cutoff point that delineates regions
where the test statistic is improbable to reside. Consequently, any observed test
statistic beyond this critical value prompts the rejection of the null hypothesis. While
critical values can be computed, they are typically derived from statistical tables
based on the chosen significance level and degrees of freedom for convenience.
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220 10 Foreground Toolbox for Hypothesis Testing
Decision rule
An essential aspect of hypothesis testing is the decision rule, which dictates when
there is adequate evidence to support the null hypothesis (H_0). Essentially, it
establishes the criteria and guidelines under which the null hypothesis should be
rejected. This rule is employed to determine whether the null hypothesis should be
rejected by comparing the test statistic to the critical value. The decision rule for a
specific test is contingent upon three key factors: the alternative hypothesis, the test
statistic, and the significance level. This comparison yields two potential outcomes:
1. If the test statistic surpasses the critical value, the null hypothesis (H_0) is
rejected. This indicates that there is ample evidence to suggest that the
alternative hypothesis is valid.
2. Conversely, if the test statistic does not exceed the critical value, the null
hypothesis is not rejected. This signifies that there is insufficient evidence to
warrant the rejection of the null hypothesis. However, it is crucial to note that
one should refrain from asserting acceptance of the null hypothesis.
When employing the p-value and significance level (α) instead of the test statistic
and critical value, the decision rule is as follows:
•
If the significance level exceeds the p-value (α > p), the null hypothesis (H _0) is
rejected.
•
Otherwise, if the significance level is less than or equal to the p-value, the null
hypothesis is not rejected.
One-tailed and two-tailed test
Statistical significance tests can be conducted as either one-tailed or two-tailed,
representing alternative approaches. The selection between a one-tailed and a
two-tailed test hinges on the formulation of the alternative hypothesis.
Two-tailed test: A two-tailed test can detect when the population parameter differs
in either direction but has less power than a one-tailed test. When the estimated
value is greater or less than a certain value of parameter, a two-tailed test, which is
known as a non-directional hypothesis, is appropriate. In this case, the alternative
hypothesis states that the population parameter may be either greater than or less
than the hypothesized value. For example, assuming the following notation:
μ
0
the claimed or hypothesized mean of population,
μ
1
the mean of first population,
μ
2
the mean of second population,
p
0
the claimed or hypothesized proportion of population.
p
1
the proportion of first population,
p
2
the proportion of second population.
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10.2 Definitions 221
Different possibilities to write the alternative hypothesis (H
A
)intwo-tailedtest
are as follows:
H
A
: p
1
/= p
2
or p
1
− p
2
/= 0 To compare proportion of two populations
H
A
: μ
1
/= μ
2
or μ
1
− μ
2
/= 0 To compare mean of two populations
H
A
: p /= p
0
or p − p
0
/= 0 To compare proportion of a population with a fixed p
0
H
A
: μ /= μ
0
or μ − μ
0
/= 0 To compare mean of a population with a fixed μ
0
One-tailed test: A one-tailed test can detect when the population parameter differs
only in one direction (left or right), but not both. It has greater power than a two-tailed
test, but it cannot detect whether the population parameter differs in the opposite
direction. When the estimated value departs from the reference value of parameter
in only one direction, a one-tailed test, which is known as a directional hypothesis, is
appropriate. In this case, the alternative hypothesis determines whether the population
parameter differs from the hypothesized value in a specific direction. For instance,
employing the above notations for two-tailed test, we can write different possibilities
for the alternative hypothesis (
H
A
) in one-tailed test as follows:
H
A
: p
1
< p
2
or p
1
> p
2
To compare proportion of two populations
H
A
: μ
1
<μ
2
or μ
1
>μ
2
To compare mean of two populations
H
A
: p
1
< p
0
or p
1
> p
0
To compare proportion of a population with a fixed p
0
H
A
: μ
1
<μ
0
or μ
1
>μ
0
To compare mean of a population with a fixed μ
0
See Fig. 10.1.
One-tailed or two-tailed Tests
Because the true difference rarely occurs in only one side of the distribution, in the
vast majority of tests, there is no reason to calculate the p-value with one-tailed test,
and performing a two-tailed test is recommended. But if the alternative hypothesis
(
H
A
) is limited to one side of the distribution, one-tailed test is preferred. The
values corresponding to α level for the one-tailed test are slightly lower than the
corresponding values for the two-tailed test. For example, the corresponding value
for 0.05 in the one-tailed test is 1.64 (1.64 times of the standard deviation away
from the mean) and in the two-tailed test is 1.96 (1.96 times of the standard
deviation away from the mean), which can sometimes change the conclusion of the
statistical test.
Tip
You must not perform first a two-tailed test, obtain non-significant result, and
then try a one-tailed test to see if that is statistically significant.
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222 10 Foreground Toolbox for Hypothesis Testing
Fig. 10.1 One-tailed and two-tailed distributions
The two-tailed tests are the default choice, unless there is very good evidence
to use one-tailed. Because in most studies, the researcher is interested to determine
whether there is a positive effect or a negative effect and the results in either direction
(positive or negative) are important for him.
One-tailed tests are rarely appropriate. The use of a one-tailed test is only
permissible if there are many unexpected one-direction changes. However, in case
of doubt, the use of one-tailed p is not recommended. However, if there is a
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10.3 The Steps for Testing Hypotheses 223
Fig. 10.2 Steps for testing
Hypothesis
tendency to perform a one-tailed test, such a decision should be made at the
beginning of the study before you look at the data.
There are some cases where one-tailed tests are not only a valid option, but truly
are a requirement (Fig.
10.2).
10.3 The Steps for Testing Hypotheses
Various steps have been proposed for testing hypothesis. In this book, seven steps of
performing the hypothesis test proposed by Niemann and Pearson will be explained.
•
Formulate the null and alternative Hypotheses (H
0
and H
A
).
•
Specify the significant level (α) and sample size (n).
•
State the test statistic and the rejection region.
•
Determine the decision rule and critical value.
•
Calculate estimations from random samples, such as means, proportions, and
standard deviations.
•
Calculate the test statistic.
•
Decision, judgment, and conclusion.
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224 10 Foreground Toolbox for Hypothesis Testing
10.4 Applications of Statistical Tests
The general applications of the common tests (Z, t, F and χ
2
) will be briefly described
here.
The Z Tes t
When the variance of population from which the sample is taken, is known, and
the known value of variance is used in the test statistic, the Z test is applied for the
following:
•
Comparing the mean of a population with a fixed number (specified in null
hypothesis).
•
Comparing the mean of two populations.
•
Comparison of the mean difference between two populations with a fixed number
(specified in null hypothesis).
•
This test should also be used when the goal is to compare proportions:
•
Comparison of proportion in population with a fixed ratio (specified in null
hypothesis).
•
Comparing the proportion in two populations.
•
Comparison of correlation coefficient with a fixed number (specified in null
hypothesis).
The t-tests
This test will be useful if the variance of the r eference population from which the
sample was taken is unknown or the sample size i s small (less than 30), and the
estimation of sample variance is used in the calculation of test statistics:
•
Comparing the mean of a population with a fixed number (specified in null
hypothesis).
•
Comparing the mean of two populations (independent, paired, matched).
•
Comparison of the mean difference between two populations with a fixed number
(specified in null hypothesis).
•
Comparison of regression coefficient (β ) with a fixed number (specified in null
hypothesis).
The F tests
In this book, the F test is used only for the following two cases.
•
Comparing the variance of two populations.
•
Comparing the mean in more than two populations.
The χ
2
test
The Chi-squared (χ
2
) test is used for the following two cases.
•
Relationship or association between the two quantitative variables in contingency
tables.
•
Goodness of fit of an observed distribution to a theoretical one.
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10.5 Common Statistical Tables 225
10.5 Common Statistical Tables
To conclude and judge about any statistical test and to find critical values, one
should refer to the statistical distributions tables. Of course, in analyses performed
by statistical software, test statistic and p-values are presented, but they still do not
provide a critical value. In order to give readers easy access, the distribution tables
of z, t, f , and
χ
2
are given in the appendix only for some common confidence levels
and limited degrees of freedom.
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Chapter 11
Hypothesis Testing
11.1 Introduction
In the previous chapter, a foreground toolbox of statistical tests including the
concepts, definitions, and stages of performing significant tests was explained in
detail so that we can deal directly with statistical tests i n this chapter. Thus, these
two chapters are complementary to each other, and wherever you need to
understand more, do not hesitate to refer to Chap.
10.
For each part of this section, an example is provided first to clarify what the
researcher means by the test and what s/he wants to judge and decide. Then, we
calculate 95% interval estimate (95% CI) and judge and conclude about it. We then
execute the hypothesis test for the same example to reveal the tight relationship
between the significant tests and the confidence interval.
Before explaining the hypothesis tests, the common symbols and Greek letters
that are often used in this chapter are introduced.
μ The actual proportion of the population p The actual proportion of the population
μ
0
Hypothesized mean of the population in
the null hypothesis
p
0
Hypothesized proportion of the population
in the null hypothesis
μ
1
The actual mean in the first population p
1
The proportion of the property in the first
population
μ
2
The actual mean in the second
population
p
2
The proportion of the property in the
second population
x Estimated sample mean n Total sample size
x
1
The estimated mean in the first
population
n
1
Thefirstsamplesize
x
2
The estimated mean in the second
population
n
2
The second sample size
σ
2
1
The known variance in the first
population
S
2
1
The estimated variance from the first
sample
σ
2
2
The known variance in the second
population
S
2
2
The estimated variance from the second
sample
p The average of two proportions S
2
p
The pooled variance of two samples
© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024
S. H. Saneii and H. Doosti, Practical Biostatistics for Medical and Health Sciences,
https://doi.org/10.1007/978-981-97-3083-4_11
227
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228 11 Hypothesis Testing
All tests in this book are done with two-tailed test. That is, the error will be on
both sides
α
2
and 1 −
α
2
.
11.2 Comparing the Mean of a Population with a Fixed
Val ue
The variance of population is Known
Example 11.1 Plasma volume in a sample of 9 from a population with a variance
of 1.0 liter is: 3.11–3.25–3.49–3.26–2.76–3.37–2.86–2.75–3.05. Can the plasma
volume of the population from which this sample is taken be equal to 3.5?
Solution: The available information show that μ
0
= 3.5, n = 9, x = 3.10,σ
2
=
0.1,σ = 0.316
and the estimated standard error of the mean can be calculated
Se =
σ
√
n
=
0.316
√
9
= 0.105. Since the variancof the population is known, we use
the standard normal distribution (
z
0.975
= 1.96.) and the 95% confidence interval is
equal to:
95% CI = x ± z
0.975
× Se = 3.10 ± 1.96 × 0.105
= 3.10 ± 0.206 =
(
2.89 to 3.31
)
The 95% confidence interval does not contain the hypothesized and fixed value
of 3.5, so there is no evidence that the plasma volume of the population from which
this sample was taken should be equal to 3.5.
Using the Test:
The null and alternative hypotheses are:
H
0
: μ = 3.5
H
A
: μ /= 3.5
Sample size is 9 and significant level is 0.05.
Test statistic is z =
(
x−μ
0
)
σ
√
n
.
Critical value is z
0.975
= 1.96. If the test statistic is greater than the critical value
of 1.96, then we reject the null hypothesis (
H
0
).
According to sample data z =
(
x−μ
0
)
σ
√
n
=
(
3.10−3.5
)
0.316
√
9
=−3.80.
Since
|
−3.80
|
> 1.96
, then we reject the null hypothesis (H
0
).
We cannot assume the mean plasma volume of population equal to 3.5. That is, the
difference between mean of the plasma volume and the value and 3.5 is significant.
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