t-test Parametric test, Biostatics and Research Methodology

1. Parametric Test: t-test/Student t-test
Ms. Nigar K.Mujawar
Assistant Professor,
Shri.Balasaheb Mane Shikshan Prasarak Mandal Ambap
Womens College of Pharmacy, Peth-Vadgaon,
Kolhapur, MH, INDIA.
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2. Parametric Test:
A parametric test is a statistical method used to analyze data that assumes specific characteristics about
the population distribution, typically that it follows a normal distribution. Used when sample size is
small (i.e. less than 30) and when population standard deviation is not available.
These tests often involve making inferences about population parameters, such as means or
variances.
Examples of parametric tests include:
1. t-tests
2. Z-test
3. F-test
4. Analysis of variance (ANOVA), and
5. Linear regression
6. Correlation Analysis
7. Parametric Survival Analysis
8. Parametric Bayesian Methods
9. Parametric Classification and Regression Trees (CART)
They are commonly used when certain assumptions about the data can be met, such as when the
data is continuous and normally distributed. If the assumptions are not met, non-parametric tests, which
make fewer assumptions about the population distribution, may be more appropriate.
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3. T-test:
A T-test is a statistical hypothesis test used to determine if there is a significant
difference between the means of two groups. It's commonly used when you have
two sets of data and want to compare their means to see if they are significantly
different from each other.
There are several types of T-tests:
1. Independent samples T-test:
Used when you want to compare the means of two independent groups to
determine if they are significantly different from each other.
2. Paired samples T-test:
Used when you want to compare the means of two related groups, such as before
and after measurements from the same individuals.
3. One-sample T-test:
Used when you want to compare the mean of a sample to a known value or
hypothesized population mean. 3

4. • The T-test calculates a T-value based on the means and standard deviations of
the samples, as well as their sample sizes.
• This T-value is then compared to a critical value (A critical value is the threshold used to
determine whether to reject the null hypothesis in hypothesis testing, based on the chosen significance level and
the distribution of the test statistic.) from the T-distribution to determine if the difference
between the means is statistically significant.
• If the T-value exceeds the critical value, then there is evidence to reject the null
hypothesis and conclude that there is a significant difference between the
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5. 1. Sample T-test:
• This term generally refers to any T-test that involves comparing the means of
two groups, but it doesn't specify whether the groups are independent or
related (paired).
• It's a broad term encompassing both independent samples T-tests and paired
samples T-tests.
2. Pooled T-test:
• In the context of independent samples T-tests, a pooled T-test assumes that
the variances of the two groups being compared are equal.
• The pooled variance is calculated by combining the variances of the two
groups, weighting each variance by its degrees of freedom.
• This approach can increase the precision of the estimated standard error and
thus the power of the test.
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6. 3. Unpaired T-test:
• This is another term for an independent samples T-test.
• It's called "unpaired" because the observations in one group are not paired
or matched with the observations in the other group.
• Each observation in one group is considered independent of the
observations in the other group.
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Purpose: Compares means of two independent groups to see if they're significantly
different.
Steps:
1. Define null (no difference) and alternative (difference exists) hypotheses.
2. Collect data from two separate groups.
3. Check if data in each group follows a normal distribution and if variances are similar.
4. Calculate t-statistic using group means, pooled standard deviation, and sample sizes.
5. Determine degrees of freedom(the maximum number of logically
independent values, which may vary in a data sample)
6. Compare calculated t-statistic to critical value or calculate p-value.
7. Interpret results: Reject null hypothesis if p-value is less than significance level.
Outcome:
- If p-value is low (less than significance level), there's evidence to suggest a significant
difference between group means.
- If p-value is high, there's no significant difference detected.

8. 4. Paired T-test:
• This refers to a T-test where each data point in one group is paired or
matched with a data point in the other group.
• This pairing is usually based on some natural pairing, such as before-
and-after measurements on the same individuals, or matched pairs in
a study.
• The paired T-test is used to compare the means of these paired
observations to determine if there is a significant difference between
them.
In summary, "Sample T-test" is a broad term referring to any T-test
comparing the means of two groups, while "Pooled, Unpaired, and
Paired" describe specific variations or conditions within the context of
independent samples T-tests or paired samples T-tests. 8

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Applications of t-test:
1. Comparing Means: Assess if there's a significant difference between two groups'
averages.
2. Before and After Studies: Evaluate changes in a variable before and after an
intervention.
3. Quality Control: Ensure products meet standards by comparing measurements
to set values.
4. Market Research: Determine differences in responses between different
consumer groups.
5. Psychological Studies: Compare mean scores of different experimental groups.
6. Biological Studies: Analyze differences in measurements between groups of
organisms.
7. Economic Analysis: Compare economic indicators between populations or
assess policy impacts.

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Assumptions of t-test
A.Independent Samples t-test:
1. Independent Observations: The observations in each group are independent
of each other. This means that the score of one individual does not affect or
influence the score of another individual.
2. Normal Distribution: The data in each group should follow a normal
distribution. However, the t-test is robust to violations of normality when
sample sizes are large (typically considered as n > 30). If sample sizes are small,
a violation of normality can affect the validity of the t-test.
3. Homogeneity of Variance: The variances of the two groups should be equal
(homogeneity of variances). This means that the spread of scores within each
group should be similar.
4. Population standard deviation is not known.
5. Samples are random and independent.

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B.Paired Samples t-test:
1. Independence of Differences: The differences between paired observations
should be independent of each other. For example, if you're comparing pre-
test and post-test scores of the same individuals, the change in one individual's
score should not be influenced by the change in another individual's score.
2. Normal Distribution of Differences: The differences between paired
observations should be approximately normally distributed.
3. Homogeneity of Differences (Optional): While not always necessary, if the
sample size is small, it's advisable for the differences between paired
observations to have a similar spread (homogeneity of differences).

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Formula in t test:

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One sample t-test formula

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Two sample t-test formula

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