This Slides presents different types of Parametric Test- like T-test, Parametric Test, Assumption of Parametric Test, Paired T Test, One Sample T Test, ANOVA, ANCOVA, Regression, Two Way ANOVA, Repeated Measure ANOVA, Multiple Regression

1. PARAMETRIC
TESTS
DR. VIKRAMJIT SINGH

2. PARAMETRIC
TESTS
DR. VIKRAMJIT SINGH

3. BASIC ASSUMPTIONS FOR
PARAMETRIC TESTS:
2.1
Assumption 1: Normality
Parametric tests assume that each group is roughly
normally distributed. The data in each group should
follow a roughly normal distribution. While the t-test is
robust to deviations from normality, it performs best
when the data is reasonably close to a normal
distribution.

4. BASIC ASSUMPTIONS FOR
PARAMETRIC TESTS:
2.2
Assumption 2: Equal Variance/Homogeneity of Variance
Parametric tests assume that the variance of each group
is roughly equal. The variances of the two groups being
compared should be approximately equal. This
assumption is known as homoscedasticity.

5. BASIC ASSUMPTIONS FOR
PARAMETRIC TESTS:
2.3
Assumption 3: Independence
Parametric tests assume that the observations in each
group are independent of observations in every other
group. Random assignments adds to this assumption. The
observations in each group must be independent of each
other. In other words, the values in one group should not
be related to the values in the other group.

6. BASIC ASSUMPTIONS FOR
PARAMETRIC TESTS:
2.4,5,6
Assumption 4: No Outliers
Parametric tests assume that there are no extreme
outliers in any group that could adversely affect the
results of the test.
Assumption 5: Ratio and Interval Scale
The collected data from each group has to be done on
ratio and interval scale.
Assumption 6: Usually a Large Sample
The sample size generally is a large sample, usually
more than 30.

7. PARAMETRIC TESTS:
STUDENT'S T-TEST
3.1
- Assumption: Assumes that the data is normally
distributed and that the variances of the groups being
compared are approximately equal. It is a parametric
statistical test used to compare the means of two
independent groups.
- Purpose: Used to compare means between two groups.
- Examples:
- Comparing the mean test scores of students who
received a new teaching method to those who received
the traditional method.

8. PARAMETRIC TESTS:
STUDENT'S T-TEST
3.1
- Examples:
- Comparing the average income of two different cities.
- Comparing the mean test scores of two groups, Group
A (n1 = 30) and Group B (n2 = 25).
-Drug Efficacy: You want to know if a new drug is more
effective in reducing blood pressure compared to an
existing drug. You collect blood pressure measurements
from 20 patients before and after using each drug.

9. PARAMETRIC TESTS:
PAIRED T-TEST
3.2
- Assumption: Similar to the Student's t-test, it
assumes normally distributed data, and the differences
between paired observations are approximately normally
distributed.
- Purpose: Used to compare means of two related
groups, such as pre-test and post-test measurements.

10. PARAMETRIC TESTS:
PAIRED T-TEST
3.2
- Example: Comparing the mean blood pressure before
and after a treatment in the same group of patients.
-Suppose you want to determine if a new diet plan is
effective in reducing participants' weight. You collect
weight measurements from 20 participants before and
after following the diet plan.

11. PARAMETRIC TESTS:
ONE-SAMPLE T-TEST
3.3
Use this test when you have a single group of data and
want to determine if the mean of that group is
significantly different from a known or hypothesized
population mean.
Assumptions: Independence of observations, normality of
data.
Example: Suppose you work for a manufacturing company
that claims their products have an average lifespan of
1000 hours.

12. PARAMETRIC TESTS:
ANALYSIS OF VARIANCE (ANOVA)
3.4
ANOVA, or Analysis of Variance, is a statistical technique used to
compare means among three or more groups or populations. ANOVA
assesses whether there are statistically significant differences in
means and can help identify which specific group(s) differ from the
others. There are several types of ANOVA, including one-way
ANOVA, two-way ANOVA, and repeated measures ANOVA.
- Assumption: Assumes that the data is normally distributed and
that the variances are approximately equal across all groups.
- Purpose: Used to compare means among three or more groups.

13. PARAMETRIC TESTS:
ANALYSIS OF VARIANCE (ANOVA)
3.4 - Examples:
- Comparing the mean exam scores of students in three different
schools.
- Analyzing the effect of different doses of a drug on patients' pain
levels.
- Example: Assessing if there are differences in the mean scores of a
test among three different age groups.
- Example: Comparing the mean exam scores of students in three
different schools (School A, B, and C).
- Example: Assessing if there are differences in the mean scores of a
test among three different age groups (Group 1, Group 2, Group 3).

14. PARAMETRIC TESTS:
ANALYSIS OF VARIANCE (ANOVA)
3.4
Hypotheses:
- Null Hypothesis (H0): There is no significant difference in
mean exam scores among the three schools .
- Alternative Hypothesis (H1): There is a significant difference
in mean exam scores among at least one pair of schools.
You perform a one-way ANOVA to determine if there are
statistically significant differences in mean scores among the
schools.

15. PARAMETRIC TESTS:
TWO-WAY ANOVA
3.5
- Assumption: It extends the one-way ANOVA by
allowing for the assessment of the effects of two
independent variables.
- Purpose: Used when there are two independent
variables to assess their main effects and potential
interaction.

16. PARAMETRIC TESTS:
TWO-WAY ANOVA
3.5
- Example: Analyzing whether the performance of
students is influenced by both teaching method and
gender.
- Formula: Extends ANOVA to assess the effects of two
independent variables.
Use two-way ANOVA when you have two independent
variables (factors) and you want to assess their main
effects and the potential interaction between them.

17. PARAMETRIC TESTS:
TWO-WAY ANOVA
3.5 Suppose you are studying student performance and want to analyse
whether teaching method (Method 1 and Method 2) and gender (Male and
Female) have any influence on exam scores. You collect the following data:
- Method 1, Male (Group 1): Mean score = 85, Sample variance = 16, Sample
size (n1) = 30
- Method 1, Female (Group 2): Mean score = 78, Sample variance = 14,
Sample size (n2) = 25
- Method 2, Male (Group 3): Mean score = 92, Sample variance = 18, Sample
size (n3) = 35
- Method 2, Female (Group 4): Mean score = 88, Sample variance = 20,
Sample size (n4) = 28

18. PARAMETRIC TESTS:
REPEATED MEASURES ANOVA
3.6
Use repeated measures ANOVA when you have one group of subjects
measured under multiple conditions or time points. It assesses the
impact of a within-subjects factor.
Example Case:
Suppose you are evaluating the effect of a new drug on patients'
pain levels. You measure their pain levels before taking the drug
(Time 1), immediately after taking the drug (Time 2), and one hour
later (Time 3). You collect data for a single group of patients:

19. PARAMETRIC TESTS:
REPEATED MEASURES ANOVA
3.6 - Time 1 (Baseline): Mean pain level = 5, Sample variance = 1, Sample size (n1) =
20
- Time 2 (Immediate): Mean pain level = 3, Sample variance = 1.5, Sample size
(n2) = 20
- Time 3 (One Hour): Mean pain level = 2, Sample variance = 1.2, Sample size
(n3) = 20
Hypotheses:
- Null Hypothesis (H0): There is no significant difference in mean
pain levels across the three time points.
- Alternative Hypothesis (H1): There is a significant difference in
mean pain levels across at least one pair of time points.

20. PARAMETRIC TESTS:
REPEATED MEASURES ANOVA
3.6 You perform a repeated measures ANOVA to examine the impact of
time on pain levels.
ANOVA is a powerful tool for comparing means in various research
settings, allowing you to determine if there are significant
differences among multiple groups or conditions.
ANOVA (Analysis of Variance) is a statistical technique used to
compare means among three or more groups or populations. Before
conducting an ANOVA, it's essential to understand its assumptions,
formulas, and examples. Let's go through these aspects.

21. PARAMETRIC TESTS:
ANALYSIS OF COVARIANCE (ANCOVA)
3.7
- Assumption: It combines aspects of ANOVA and regression,
assuming normally distributed data, equal variances, and the
ability to control for covariates.
- Purpose: Used to compare means among groups while
controlling for the influence of one or more continuous
covariates.
- Example: Assessing if there is a significant difference in test
scores among three schools while controlling for students'
socioeconomic status.

22. PARAMETRIC TESTS:
ANALYSIS OF COVARIANCE (ANCOVA)
3.7 ANCOVA (Analysis of Covariance) is a statistical technique used to compare means
among groups while controlling for the influence of one or more continuous
covariates. ANCOVA combines elements of both analysis of variance (ANOVA) and
regression analysis. Let's delve into the assumptions, formula, and example cases for
ANCOVA.
Assumptions of ANCOVA:
1. Independence: Observations within each group must be independent of each other.
2. Homogeneity of Variances (Homoscedasticity): The variances of the groups being
compared should be approximately equal.
3. Normality: The residuals (the differences between the observed values and the
predicted values) should follow a roughly normal distribution.
4. Linearity: The relationship between the covariate(s) and the dependent variable
should be linear.

23. PARAMETRIC TESTS:
ANALYSIS OF COVARIANCE (ANCOVA)
3.7 Example 1 - ANCOVA with One Covariate:
Scenario: You want to assess whether there is a significant difference in
the mean exam scores of students from three different schools (School A,
School B, and School C), while controlling for the influence of students'
socioeconomic status (SES), which is a continuous covariate. You collect
the following data:
- School A (Group 1): Mean score = 85, Sample variance = 16, Sample size (n1) = 30
- School B (Group 2): Mean score = 78, Sample variance = 14, Sample size (n2) = 25
- School C (Group 3): Mean score = 92, Sample variance = 18, Sample size (n3) = 35
You also collect SES data for each student.

24. PARAMETRIC TESTS:
ANALYSIS OF COVARIANCE (ANCOVA)
3.7 Hypotheses:
- Null Hypothesis (H0): There is no significant difference in mean
exam scores among the three schools, after controlling for SES.
- Alternative Hypothesis (H1): There is a significant difference in
mean exam scores among at least one pair of schools, after controlling
for SES.
You perform an ANCOVA to determine if there are statistically
significant differences in mean scores among the schools while
accounting for SES as a covariate.

25. PARAMETRIC TESTS:
ANALYSIS OF COVARIANCE (ANCOVA)
3.7 Example 2 - ANCOVA with Multiple Covariates:
Scenario: You are evaluating the effect of three different diets on
weight loss. You collect weight loss data for three groups of
participants (Diet A, Diet B, Diet C) and want to control for the
influence of both age and initial body mass index (BMI), which are
continuous covariates. You collect the following data:
- Diet A (Group 1): Mean weight loss = 5 kg, Sample variance = 4, Sample size (n1) = 30
- Diet B (Group 2): Mean weight loss = 6 kg, Sample variance = 3.5, Sample size (n2) =
25
- Diet C (Group 3): Mean weight loss = 4 kg, Sample variance = 4.2, Sample size (n3) =
35
You also collect age and initial BMI data for each participant.

26. PARAMETRIC TESTS:
ANALYSIS OF COVARIANCE (ANCOVA)
3.7 Hypotheses:
- Null Hypothesis (H0): There is no significant difference in mean weight
loss among the three diets, after controlling for age and initial BMI.
- Alternative Hypothesis (H1): There is a significant difference in mean
weight loss among at least one pair of diets, after controlling for age and
initial BMI.
You perform an ANCOVA with multiple covariates (age and initial BMI) to
assess whether there are statistically significant differences in mean weight
loss among the diets while considering the covariates' influence.
ANCOVA allows you to examine group differences while taking into account
the potential impact of continuous covariates, making it a valuable tool for
research when controlling for confounding variables.

27. PARAMETRIC TESTS:
MULTIPLE REGRESSION
3.8
Multiple Regression is a statistical technique used to analyze
the relationship between a dependent variable (or target)
and two or more independent variables (or predictors) by
estimating the coefficients of a linear equation. It helps us
understand how changes in the predictor variables are
associated with changes in the dependent variable. Let's
explore the assumptions, formula, and example cases for
multiple regression.

28. PARAMETRIC TESTS:
MULTIPLE REGRESSION
3.8 Assumptions of Multiple Regression:
1. Linearity: There should be a linear relationship between the predictors and the
dependent variable. You can check this assumption using scatterplots or residual plots.
2. Independence of Errors: The errors (residuals) should be independent of each other. In
other words, the value of the error for one observation should not depend on the value of
the error for another observation.
3. Homoscedasticity: The variance of the errors should be approximately constant across
all levels of the predictors. You can check this assumption by examining residual plots.
4. Normality of Residuals: The residuals should be normally distributed. You can assess
this assumption using normal probability plots or statistical tests like the Shapiro-Wilk
test.
5. No or Little Multicollinearity: The predictor variables should not be highly correlated
with each other. High multicollinearity can make it challenging to interpret the individual
coefficients of the predictors.

29. PARAMETRIC TESTS:
MULTIPLE REGRESSION
3.8 Multiple Regression Formula:
The multiple regression model can be represented as:
The goal is to estimate the coefficients that minimize the
sum of squared errors (residuals) between the predicted
values and the actual values of the dependent variable.

30. PARAMETRIC TESTS:
MULTIPLE REGRESSION
3.8
Example Cases:
Example 1 - Predicting House Prices:
Scenario: You want to predict house prices based on several
independent variables, including square footage, number of bedrooms,
number of bathrooms, and neighborhood safety score. You collect data
for 100 houses.
You build a multiple regression model as follows:
House Price = β0 + β1 Square Footage + β2 Bedrooms + β3Bathrooms
+ β4SafetyScore + epsilon
You estimate the coefficients and assess the model's performance in
predicting house prices.

31. PARAMETRIC TESTS:
MULTIPLE REGRESSION
3.8
Example 2 - Employee Performance:
Scenario: You work in HR and want to understand the factors that
influence employee performance scores. You collect data on several
potential predictors, such as years of experience, education level, and
the number of training hours, for 200 employees.
You build a multiple regression model as follows:
Employee Performance = β0 + β1 Experience + β2 Education + β3
Training Hours + epsilon
You estimate the coefficients to identify which factors have a
significant impact on employee performance.

32. PARAMETRIC TESTS:
MULTIPLE REGRESSION
3.8
Multiple regression is a powerful tool for exploring and
quantifying relationships between variables. It allows you to
make predictions, test hypotheses, and uncover insights in
various fields such as economics, social sciences, and data
analysis. However, it's crucial to validate the assumptions and
interpret the results carefully to draw meaningful conclusions.

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