Parametric Tests
By: Saumyadeepta Sen
Tags: Inferential Statistics, Hypothesis Testing, Mathematics, Statistics, Statistical Analysis

Parametric Tests are used for the following cases:

Quantitative Data Continuous variable When data is measured on approximate interval or ratio scales of measurement. When data should follow Normal Distribution

Types Of Parametric tests

t-test(n< 30), which is further classified into 1-sample and 2-sample Anova(Analysis of Variance)- One way Anova, Two way Anova Pearson’s r Correlation Z-test for large samples(n> 30)

Student’s T-Test

This test was developed by Prof. W.S.Gossett in 1908, who published statistical papers under the pen name of ‘Student’. Thus the test is known as Student’s t-test.

Indications for the test:

1.When samples are small

2.Population Variances are not known.

Uses:

1. Compare two means of small independent samples

2.Compare Sample mean and Population mean

3.Compare two proportions of small independent samples.

Assumptions:

1.Samples are randomly selected

2.Data utilised is quantitative

3.Variable follow Normal distribution

4.Sample variance are mostly same in both the groups under study

5.Samples are small, mostly lower than 30.

A t-test compares the difference between two means of different groups to determine whether the difference is statistically significant.

Student’s t-test for different purposes:

t-test for one-sample t-test for unpaired two samples t-test for paired two samples

ONE SAMPLE T-TEST

When we compare the mean of a single group of observations with a specified value. In one sample t-test, we know the population mean. We draw a random sample from the population and then compare the sample mean with the population mean and make a statistical decision as to whether or not the sample mean is different from the population.

Calculations:

Now we compare calculated value with table value at certain level of significance (generally 5% or 1%)

If absolute value of ‘t’ obtained is greater than table value then reject the null hypothesis and if it is less than table value, the null hypothesis may be accepted.

Unpaired Two sample t- test

Compare two means of two samples

Two independent random samples come from the normal populations having unknown or same variance

We test the null hypothesis, that the two population means are same against an appropriate one sided or two sided alternative hypothesis.

Assumptions:

1. The samples are random & independent of each other.

2. The distribution of dependent variable is normal.

3. The variances are equal in both the groups.

Calculations:

Here , Xbar_₁ and Xbar_₂ are the sample means, s² is the pooled sample variance, n1 and n2 are the sample sizes and t is a Student-t quantile with n1 + n2–2 degrees of freedom.

PAIRED TWO-SAMPLES T-TEST

Used when we have paired data of observations from one sample only, when each individual gives a pair of observations.

Same individuals are studied more than once in different circumstances- measurements made on the same people before and after interventions.

Assumptions:

1.The outcome variable should be continuous.

2.The difference between pre-post measurements should be normally distributed.

Calculations:

Here,

d=difference between two samples X1 and X2,

dbar= Mean of d,

Sd= Standard Deviation of the difference,

n=Sample size.

Now , how do we compare more than two groups means ??

Instead of using a series of individual comparisons as we do in t-test, we examine the differences among the groups through an analysis that considers the variation among all groups at once. • i.e. ANALYSIS OF VARIANCE

Analysis of Variance(ANOVA)

Given by Sir Ronald Fisher

The principle aim of statistical models is to explain the variation in measurements.

The statistical model involving a test of significance of the difference in mean values of the variable between two groups is the student’s,’t’ test. If there are more than two groups, the appropriate statistical model is Analysis of Variance (ANOVA)

Assumptions:

1.Sample population can be easily approximated to normal distribution.

2.All populations have same Standard Deviation.

3. Individuals in population are selected randomly.

4.Independent samples.

ANOVA compares variance by means of a simple ratio, called F-Ratio ,

F= (Variance between groups) / (Variance within groups) .

, . The resulting F statistics is then compared with critical value of F (critic) , obtained from F tables in much the same way as was done with ‘t’.

is then compared with critical value of , obtained from in much the same way as was done with ‘t’. If the calculated value exceeds the critical value for the appropriate level of α, the null hypothesis will be rejected.

A F test is therefore a test of the Ratio of Variances. F Tests can also be used on their own, independently of the ANOVA technique, to test hypothesis about variances.

In ANOVA, the F test is used to establish whether a statistically significant difference exists in the data being tested.

ANOVA is further divided into:

*One way ANOVA

*Two way ANOVA

One Way ANOVA

If the various experimental groups differ in terms of only one factor at a time- a One way ANOVA is used.

e.g. A study to assess the effectiveness of four different antibiotics on S Sanguis.

Two Way ANOVA

If the various groups differ in terms of two or more factors at a time, then a Two way ANOVA is performed.

e.g. A study to assess the effectiveness of four different antibiotics on S Sanguis in three different age groups.

Pearson’s Correlation Coefficient

Correlation is a technique for investigating the relationship between two quantitative, continuous variables

Pearson’s Correlation Coefficient(r) is a measure of the strength of the association between the two variables.

Assumptions:

1. Subjects selected for study with pair of values of X & Y are chosen with random sampling procedure.

2. Both X & Y variables are continuous

3. Both variables X & Y are assumed to follow normal distribution.

Steps:

*The first step in studying the relationship between two continuous variables is to draw a scatter plot of the variables to check for linearity.

*The correlation coefficient should not be calculated of the relationship is not linear.

*For correlation only purposes, it does not matter on which axis the variables are plotted.

However, conventionally, the independent variable is plotted on X axis and dependent variable on Y-axis.

The nearer the scatter of points is to a straight line, the higher the strength of association between the variables.

Types of Correlation:

Perfect Positive Correlation r=+1

Partial Positive Correlation 0<r<+1

Perfect negative correlation r=-1

Partial negative correlation 0>r>-1

No Correlation r=0

Z- Test

This test is used for testing significance difference between two means (n>30).

Assumptions:

1.The sample must be randomly selected.

2.Data must be quantitative.

3.Samples should be larger than 30.

4.Data should follow normal distribution.

5.Sample variances should be almost the same in both the groups of study.

Note:

• If the SD of the populations is known, a Z test can be applied even if the sample is smaller than 30.

Indications:

*To compare sample mean with population mean.

*To compare two sample means.

*To compare sample proportion with population proportion.

*To compare two sample proportions.

Steps:

1. Defining the problem

2. Stating the null hypothesis (H0) against the alternate hypothesis (H1)

3. Finding Z value, Z= (Observed mean)/(Mean Standard Error).

4. Fixing the level of significance

5. Comparing calculated Z value with the value in Z table at corresponding degree significance level.

If the observed Z value is greater than theoritical Z value, Z is significant, we reject the null hypothesis and accept the alternate hypothesis.

Z- PROPORTIONALITY TEST

Used for testing the significant difference between two proportions.

Calculations:

One tailed and Two tailed Z tests

Z values on each side of mean are calculated as +Z or as -Z.

• A result larger than difference between sample mean will give +Z and result smaller than the difference between mean will give -Z.

E.g. for two tailed test:

In a test of significance, when one wants to determine whether the mean IQ of malnourished children is different from that of well nourished and does not specify higher or lower, the P value of an experiment group includes both sides of extreme results at both ends of scale, and the test is called two tailed test.

E.g. for single tailed:

In a test of significance when one wants to know specifically whether a result is larger or smaller than what occur by chance, the significant level or P value will apply to relative end only e.g. if we want to know if the malnourished have lesser mean IQ than the well nourished, the result will lie at one end (tail)of the distribution, and the test is called single tailed test.

Conclusion

Tests of significance play an important role in conveying the results of any research & thus the choice of an appropriate statistical test is very important as it decides the fate of outcome of the study.

as it decides the fate of outcome of the study. Hence the emphasis placed on tests of significance in clinical research must be tempered with an understanding that they are tools for analyzing data & should never be used as a substitute for knowledgeable interpretation of outcomes.

Link to Part 1(Parametric and Non-parametric tests for comparing two or more groups) :

https://medium.com/the-owl/parametric-and-non-parametric-tests-for-comparing-two-or-more-groups-part-1-e24cbbbe4321