Provide Information Regarding Statistics & Econometrics : Analysis of Covariance lecture

Analysis of Covariance

ANCOVA

lecture - 20

What is covariate?

The uncontrolled continuous measurable variable, that runs along with the main variable and is suspected to influence the main variable (Dependent variable) but not influence the other independent variables included in model is called covariate or concomitant variable.

Let us there is a desire to assess how different teaching methods (consider treatments) effect student’s score (consider response variable). Before conducting the experiment, it is known the every student has a different foundation of knowledge. While some students have prior knowledge about the subject, other do not. The students are divided into two groups, one of which is already familiar with the subject, while the other is not. In this case, prior knowledge to subject is a covariate.

When ANOVA Model is applicable?

The ANOVA model is used to check the influence of categorical variable on the dependent variable.

Example:

A screening test company conducted a test for a particular job. The Test score of individuals is depend on education levels (12 years of schooling, 14 years of schooling, 16 years of schooling).

12 years schooling	14 years schooling	16 years schooling
50	56	60
34	30	42
77	82	56
51	49	59

When Regression Model is applicable?

The regression model is used to check the influence continuous variable on dependent variable.

Example:

In the above example, if the test score is depended on study skills and take GPA as study skill.

12 years schooling	GPA	14 years schooling	GPA	16 years schooling	GPA
50	2.50	56	2.71	60	2.75
34	2.60	30	2.50	42	3.00
77	3.20	82	2.90	56	3.25
51	3.00	49	2.56	59	2.89

What is ANCOVA?

Analysis of Covariance (ANCOVA) is a blend of analysis of variance (ANOVA)and regression. It is a powerful statistical technique used to analyze the effect of treatments on the main variable “Y” while controlling the effect of at least one continuous covariate that run along the main variable and suspected to influence the main variable.

In ANCOVA first remove the effect of covariate(s) on the dependent variable by regression method which is not the primary interest to the study. Then allowing for a more accurate assessment of the relationship between the treatments and the dependent variable. In second stage treatments are analyzing by using ANOVA. Thus, statistical method that can combine ANOVA and Regression for adjusting linear effect of covariate and make a clearer picture is called the analysis of covariance (ANCOVA).

Example:

A researcher desires to test the significance difference in the average score of public, private and semi government intuitions students. But there are certain other factors like home environment, parental support, etc. are confound variables. which influence the score or performance of students. The type of intuitions are categorical variables and deal is a treatment, while home environment, parental support are covariates.

Purpose of ANCOVA

i. In experimental designs, to control for factors which cannot be randomized but which can be measured on an interval scale.

ii. In observational designs, to remove the effects of variables which modify the relationship of the categorical independents to the interval dependent.

iii. In regression models, to fit regressions where there are both categorical and interval independents.

Assumptions for ANCOVA

Assumptions are basically the same as the ANOVA assumptions. Check that the following are true before running the test:

i. The relationship between the response variable and the covariate must be linear.

ii. The slope will be constant for all treatment groups between X and Y.

iii. The independent variables are not correlated with the error term.

iv. There will be no correlation with in a group observation and between groups observation.

v. The covariate is measured without error.

vi. The data is collected from normal distribution.

vii. All effects are additive.

Statistical Model for one way ANCOA & Analysis

(ANCOVA Model in CRD)

Let Yij denote the i^th observation of j^th treatment and Xij is the corresponding covariate on dependent variable.

The ANCOVA model in case of CRD is given by:

Analysis of ANCOVA in case of CRD

Let (Yij, Xij) be the i^th pair of observation on j^th treatment of dependent variable and covariate and each pair is replicated r times. The data may be arranged as shown in the following table:

The table of sum of squares and sum of products:

Un adjusted ANOVA:

Each treatment mean is adjusted as:

Short cut formulas to compute various sums of squares:

For Y:

Short cut formulas to compute various sums of squares:

For X:

Short cut formulas to compute various sums of squares:

For XY:

Example:

The effect of three fertilizers and the effect of shadow on the yield are given below:

Fertilizer
A		B		C
X	Y	X	Y	X	Y
3	10	4	12	1	6
2	8	3	12	2	5
1	8	3	10	3	8
2	11	5	13	1	7

Test the effect of 3 fertilizers on yield at 5 % level of significance.

Solution: We using ANCOVA because the effect shadow cannot be control by randomization and blocking.

State the null and alternative hypothesis as:

i. H0 : μA = μB = μC Vs. H1 : μA ≠ μB ≠ μC

ii. The significance level; α = 0.05

iii. The Test Statistic: ANCOVA

iv. Computation:

For Y:

For X:

For XY:

Table for the sum of squares and product:

Now to test the hypothesis about slope as:

Remarks:

The null hypothesis about slope is zero rejected. The shadow (covariate) is significant effect on the yield. The dependent variable is not free from covariate X.

Computation for adjusted ANOVA:

v. Reject H0, when F >= 4.26

vi. Remarks:

The computed F value falling in the rejection region, we have not sufficient evidence to accept . Thus, it is concluded that the effect of 3 fertilizers are not identical at 5 % significance level.

Further