3B Flashcards
Two different ways to think about the size effect
- how steep is the line
- how close are the points to the line
These two types of effect size are not entirely unrelated, but they are different questions.
How steep is the line?
Two different ways to think about the effect size
This answer questions how much Y increases with every increse in X
The steeper the regression line is, the more Y changes with every unit increase in X, so the stronger the effect
The measure of this effect size is the unstandardized regression coefficient
How close are the points to the line?
Two different ways to think about the effect size
The closer the points are to the line, the better Y can be predicted based on X, the stronger the effect
The measure of this effect is the standardized regression coefficient
why is the p-value not a measure of effect size?
The t-value and hence the p-value depends on more than just the effect size. Especially in a large sample, even very small and meaningless effects can still be significant.
what are standardized coefficients?
Standardized coefficients indicate the direction of the effect and how close the points are to the regression line.
In other words:
- How well can Y be predicted based on X
- How much do you know about Y when you know X
- How commonly do X and Y go together?
The standardized coefficient always takes a value between -1 and +1
The standardized coefficient, indicates how many standard deviations Y increases with every increase of one standard deviation in X
Meaning of the values of the standardized coefficient
between -1 and +1
- Value of +1: There is a perfect positive relation between X and Y (Y can be predicted with 100% accuracy when you know X, all points are on the line)
- Value of -1: There is a perfect negative relation between X and Y (Y can be predicted with 100% accuracy when you know X, all points are on the line)
- Value of 0: The is no relation between X and Y (the points are all over the place)
How are standardized coefficients calculated?
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Why comparing standardized coefficients?
To compare the effect on Y between different X variables. An advantage of standardized coefficients is that you an always compare them between different variables
The X-variable with the highest (absolute) standardized coefficient is the most important/accurate predictor of Y
Why is it hard, if not impossible, to compare unstandardized coefficients here?
The unstandardized coefficients cannot meaningfully be compared here because two X-variables can be measured on a completely different scale.
The unstandardized coefficients can only be compared between X-variables that have a common metric
The effect size of regression models
The explained variance can be used. This is also known as R2 or R Square, a measure for the effect size of the entire mode.
In other words, the R2 indicates the effect size of all independent variables in a regression model together.
The value of R2
between 0 and 1
R2 = 0: The model does not predict any variation in Y
R2 = 1: The model predicts all variation in Y perfectly (all points are on the regression line)
Unlike standardized coefficients, R2 values cannot be negative (after all, a model cannot explain less than nothing of the variation in Y)
Keep in mind, for % –> R2 x 100%
How is the R2 calculated?
The R2 is calculated as
R2 = sum of Square Regression / Sum of Square Total
The p-value of the R2
The R 2 also has a corresponding p-value, which indicates if the R 2 is significantly larger than 0
If this p-value is smaller than 0.05, the null-hypothesis that the model as a whole explains no variation in Y (H 0 : R 2 = 0) can be rejected
The F-test
F can be found in ANOVA SPSS-output
should know that the p-value of the R 2 follows from:
1. The R 2 itself (larger R 2 -> lower p-value)
2. The number of X-variables (more X-variables -> higher p-value)
3. The sample size (larger sample size -> lower p-value)
Don’t have to calculate on the exam!
The R2 in simple linear regression
In simple linear regression, the R2 is the squared value of the correlation R
