Korean, Edit

Chapter 19. Analysis of Variance of Regression Analysis

Higher category: 【Statistics】 Statistics Overview

1. ANOVA of simple linear regression analysis

2. ANCOVA

1. ANOVA of simple linear regression analysis 

⑴ problem situation

⑵ table of t statistic 

⑶ table of F statistic

① null hypothesis H0 : the slope of the regression line is equal to zero 

② idea : if MS of regression ＞ MS of residual, the slope of the regression line is not zero 

③ calculation

④ the reason why the degree of freedom of the regression line is 1: because there is only one regression variable

⑷ the reason why F statistic shows greater power than t statistic : p value of F statistic is smaller 

① F statistic uses more information than t statistic

② F statistics shows greater power due to the effect of larger sample sizes

2. ANCOVA (analysis of covariance) 

⑴ overview

① concept of fusing simple linear regression analysis with one-way ANOVA

② necessity : in actual problem situations, the second factor changes due to the confounding effect of one factor, which can affect the dependent variable

③ difference from two-way ANOVA: ANCOVA technique does not compete with certain ANOVA techniques. ANOVA and ANCOVA can be performed at the same time

⑵ problem situation

① independent variable: whether it is a contaminated mine area or not

② dependent variable: lead concentration in organs of the rats

③ confounding effect: age

⑶ table of results without considering age

① if the age effect is not controlled, the residuals become larger

② as the residuals increase, MS Error increases and F ratio decreases

③ as F ratio decreases, the power decreases: in other words, it is difficult to prove the significance of the treatment

⑷ assumptions

① homoscedasticity

② independency

③ normality

④ the relationship between a covariate and the dependent variable should be linear

⑤ parallelism

○ for example, when calculating the regression line for each contaminated mining area and non-contaminated area, the slopes will be the same

○ satisfying parallelism means the same thing as no interaction 

○ if parallelism is not satisfied, comparing differences for one selected value (e.g., the overall mean of age) cannot represent the entire range of covariatess

○ before ANCOVA, the interaction of age and region should be evaluated to confirm parallelism

⑸ procedure

① 1^st. confirm the correlation between age and lead concentration

② 2^nd. confirm that the interaction of age and region is not statistically significant 

③ 3^rd. calculate the regression line of lead concentration according to age 

④ 4^th. calculate two regression lines satisfying the following conditions from the regression line obtained from 3^rd step 

○ for the regression line obtained from the 3rd step, change only the y intercept while maintaining the slope of the regression line 

○ minimize the sum of squares for each independent variable’s level  

⑤ 5^th. calculate the residuals from each regression line obtained from 4^th step

⑥ 6^th. after calculating the average age for the entire group, the function value of each regression line for that value is designated as the standard value

○ the average age for the entire group is just an example and it doesn’t matter what value it is.

⑦ 7^th. mark the residuals obtained from 5th step up and down at the standard value for each treatment group

⑧ 8^th. finally, you can see that SS Error is smaller: p value is smaller

⑹ result

① correction result

② table of results before correction

③ table of results after correction: the sum of squares for Age can be calculated from the regression line

④ report example : “A preliminary analysis for parallelism showed no significant difference between the slopes of the lines for lead concentration in relation to age (age × site: F1,16 = 0.00, NS). The subsequent ANCOVA showed a significant effect of site (F1,17 = 57.329, P < 0.001) as well as a significant effect of the covariate (age) (F1,17 = 93.054, P < 0.001). Rats from the mine site had higher levels of lead than those from the control.

⑺ the reason for not comparing the y-intercept of the regression line in the contaminated mine area with that of the regression line in the other area

① situation : if parallelism is satisfied, it is much easier to compare the y-intercepts 

② comparing y-intercepts is similar to comparing sample groups with only one sample

③ because ANCOVA takes the total error terms, ANCOVA is similar to comparing sample groups with twice as many elements as the size of the given sample group 

④ therefore, performing ANCOVA has higher power than simply comparing y-intercepts

⑻ application 1. 2-factor ANCOVA 

① applying the ANCOVA technique when analyzing 2-way ANOVA

② example: when the independent variables are gender and drug treatment, the dependent variable is blood pressure, and the confounding factor is age  

⑼ application 2. if there are multiple confounding factors  

① multiple linear regression analysis is used 

② advanced regression analysis may also be used

Input: 2019.12.07 23:04