# Perform three types of t-test in Python

## Student’s t-test

• Student’s t-test or t-test is a parametric inferential statistical method used for comparing the means between two different groups (two-sample t-test) or with the specific value (one-sample t-test).
• In t-test, test statistic follows the t-distribution (type of continuous probability distribution) under the null hypothesis.
• t-distribution is first proposed by William Sealy Gosset and published under the fictitious name of “Student” in Biometrika scientific journal. Hence, t-distribution is also known as Student’s t-distribution.
• In contrast to the z-test, which requires a larger sample size (t and z distributions are similar for larger sample size, say n=30), the t-test is specially developed for the small sample size data (n ≤ 30). t-test is also applies for extremely small sample size data (n ≤ 5). t-test provides more conservative results for small sample datasets.
• t-test has three main types: One Sample t-test, two sample t-test (unpaired or independent), and paired t-test.
• In one sample t-test, the sample mean is compared with the hypothesized values, whereas in two sample t-test ( independent or unpaired t test), means of two independent groups are compared. In paired t-test, differences between the pair of dependent variables are compared.

## Types of t-test

### One Sample t-test

• One Sample t-test (single sample t-test) is used to compare the sample mean (a random sample from a population) with the specific value (hypothesized or known mean of the population).
• For example, a ball has a diameter of 5 cm and we want to check whether the average diameter of the ball from the random sample (e.g. 50 balls) picked from the production line differs from the known size.

#### Assumptions

• Dependent variable should have an approximately normal distribution (Shapiro-Wilks Test)
• Observations are independent of each other

Note: One sample t-test is relatively robust to the assumption of normality when the sample size is large (n ≥ 30)

#### Hypotheses

• Null hypothesis: Sample mean is equal to the hypothesized or known population mean
• Alternative hypothesis: Sample mean is not equal to the hypothesized or known population mean (two-tailed or two-sided)
• Alternative hypothesis: Sample mean is either greater or lesser to the hypothesized or known population mean (one-tailed or one-sided)

#### Formula

One Sample t-test formula,

#### Calculate one sample t-test in Python

Note: If you have your own dataset, you should import it as pandas dataframe. Learn how to import data using pandas

Perform one sample t-test using SciPy,

from scipy import stats as st
from bioinfokit.analys import get_data
# load dataset as pandas dataframe
df = get_data('t_one_samp').data
# output
size
0  5.739987
1  5.254042

# t test using scipy
a =  df['size'].to_numpy()
# use parameter "alternative" for two-sided or one-sided test
st.ttest_1samp(a=a, popmean=5)
# output
Ttest_1sampResult(statistic=0.36789006583267403, pvalue=0.714539654336473)


Perform one sample t-test using bioinfokit,

#### Interpretation

The p value obtained from the one sample t-test is not significant (p > 0.05), and therefore, we conclude that the average diameter of the balls in a random sample is equal to 5 cm.

### Two sample t-test (unpaired or independent t-test)

The two-sample (unpaired or independent) t-test compares the means of two independent groups, determining whether they are equal or significantly different. In two sample t-test, usually, we compute the sample means from two groups and derives the conclusion for the population’s means (unknown means) from which two groups are drawn.

For example, we have two different plant genotypes (genotype A and genotype B) and would like to compare if the yield of genotype A is significantly different from genotype B

#### Hypotheses

• Null hypothesis: Two group means are equal
• Alternative hypothesis: Two group means are different (two-tailed or two-sided)
• Alternative hypothesis: Mean of one group either greater or lesser than another group (one-tailed or one-sided)

#### Assumptions

• Observations in two groups have an approximately normal distribution (Shapiro-Wilks Test)
• Homogeneity of variances (variances are equal between treatment groups) (Levene or Bartlett Test)
• The two groups are sampled independently from each other from the same population

Note: Two sample t-test is relatively robust to the assumption of normality and homogeneity of variances when sample size is large (n ≥ 30) and there are equal number of samples (n1 = n2) in both groups.

If the sample size small and does not follow the normal distribution, you should use non-parametric Mann-Whitney U test (Wilcoxon rank sum test)

#### Formula

Two sample (independent) t-test formula,

If the variances are equal, the two sample t-test and Welch’s test (unequal variance t-test) perform equally (in terms of type I error rate) and have similar power.

#### Calculate Two sample t-test in Python

Perform two sample t-test using SciPy,

Run the code in colab

Perform two sample t-test using bioinfokit,

Run the code in colab

**Note: Even though you can perform a t-test when the sample size is unequal between two groups, it is more efficient to have an equal sample size in two groups to increase the power of the t-test.

#### Interpretation

The p value obtained from the t-test is significant (p < 0.05), and therefore, we conclude that the yield of genotype A is significantly different than genotype B.

### Paired t-test (dependent t-test)

• Paired t-test used to compare the differences between the pair of dependent variables for the same subject
• For example, we have plant variety A and would like to compare the yield of A before and after the application of some fertilizer
• Note: Paired t-test is a one sample t-test on the differences between the two dependent variables

#### Hypotheses

• Null hypothesis: There is no difference between the two dependent variables (difference=0)
• Alternative hypothesis: There is a difference between the two dependent variables (two-tailed or two-sided)
• Alternative hypothesis: Difference between two response variables either greater or lesser than zero (one-tailed or one-sided)

#### Assumptions

• Differences between the two dependent variables follows an approximately normal distribution (Shapiro-Wilks Test)
• Independent variable should have a pair of dependent variables
• Differences between the two dependent variables should not have outliers
• Observations are sampled independently from each other

#### Formula

Paired t-test formula,

Perform Paired t-test,

Run the code in colab

#### Interpretation

The p value obtained from the t-test is significant (p < 0.05), and therefore, we conclude that the yield of plant variety A significantly increased by the application of fertilizer.

Note: If you have partially paired data, you can use an independent t-test by treating two dependent variables as two different samples or drop all unpaired observations for performing paired t-test. But, both ad hoc approaches are not appropriate as it does not follow the basic requirement and may lead to biased estimate of the variance and loss of information 6.

## Sample size recommendations for t-test

• The t-test can be applied for the extremely small sample size (n = 2 to 5) provided the effect size is large and data follows the t-test assumptions. Remember, a larger sample size is preferred over small sample sizes.
• For paired t-test, it is advisable to have a high within-pair correlation (r > 0.8) to get a high statistical power (>80%) for small sample size data.
• t-test is relatively robust to the assumption of normality and homogeneity of variances when the sample size is large (n ≥ 30).

## References

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2. Kim TK, Park JH. More about the basic assumptions of t-test: normality and sample size. Korean journal of anesthesiology. 2019 Aug;72(4):331.
3. Schober P, Vetter TR. Two-sample unpaired t tests in medical research. Anesthesia & Analgesia. 2019 Oct 1;129(4):911.
4. Zabell SL. On student’s 1908 article “the probable error of a mean”. Journal of the American Statistical Association. 2008 Mar 1;103(481):1-7.
5. De Winter JC. Using the Student’s t-test with extremely small sample sizes. Practical Assessment, Research, and Evaluation. 2013;18(1):10.
6. Guo B, Yuan Y. A comparative review of methods for comparing means using partially paired data. Statistical methods in medical research. 2017 Jun;26(3):1323-40.
7. Ruxton GD. The unequal variance t-test is an underused alternative to Student’s t-test and the Mann–Whitney U test. Behavioral Ecology. 2006 Jul 1;17(4):688-90.