Lecture 1: ANOVA
Analysis of variance = single quantitative dependent variable + one or more factors (and
possibly covariates)
Assumptions:
- Normality assumption = dependent variable normally distributed in each population
a. Skewness (>2) = positive = tail towards the right and negative = tail towards
the left
b. Kurtosis (>2) = positive = more peaked than normal and vica versa
c. Z-scores above 1,96 or below 1,96 are suspicious
d. Relatively robust when observations > 15 + normally 5% are greater
than 2 z-scores
So you always look at the histogram and tests of normality > H0 = normal distribution within
population, when alpha < 0,05 this is not the case. In both normality tests here we do not
see violation of normality.
,Next up, you check skewness and kurtosis values, which both are above or below 2, so
suspicious values = more peaked and skewed towards the left.
Lastly, in the box plot we can actually see the one outlier.
If we are dealing with outliers, we also check the z-scores. When the z-score is above or
below 1,96/-1,96 then it is a suspicious outlier which is the case for group 1 seeing the
picture below.
> Take into account normally 5% is within this suspicious range, so take 5% of your
observations and see if this really differs from the amount of outliers you have gathered
based on boxplot analysis and z-scores
, - Homogeneity of variance = variance of dependent variable is equal in each
population
a. Levene test
b. Even groups > no problem
c. Largest variance in largest group > overestimation of error variance >
significance is actually lower
d. Largest variance in smallest group > underestimation of error variance >
significance is actually higher
We see in the above example that the largest standard deviation is 2* smallest standard
deviation = Levene test significant so violation of homogeneity.
However, equality of group sizes = valid use of ANOVA > probably Welch test will
therefore show the same results.
, Bonferroni > how many divided by?
Orthogonal
Orthogonal = when you * the coefficients of two contrasts and then add all of them up = 0, if
not > not orthogonal > DELETE NON SIGNIFICANT INTERACTION EFFECT
In this example all contrasts are orthogonal e.g. 1*1 + -1*-1 + 1*-1 + -1*1 = 0
Example of non orthogonal design > see above
> look at type III sum of squares for their own explained variance, because the terms
partially explain the same variance when not orthogonal > see example below
Analysis of variance = single quantitative dependent variable + one or more factors (and
possibly covariates)
Assumptions:
- Normality assumption = dependent variable normally distributed in each population
a. Skewness (>2) = positive = tail towards the right and negative = tail towards
the left
b. Kurtosis (>2) = positive = more peaked than normal and vica versa
c. Z-scores above 1,96 or below 1,96 are suspicious
d. Relatively robust when observations > 15 + normally 5% are greater
than 2 z-scores
So you always look at the histogram and tests of normality > H0 = normal distribution within
population, when alpha < 0,05 this is not the case. In both normality tests here we do not
see violation of normality.
,Next up, you check skewness and kurtosis values, which both are above or below 2, so
suspicious values = more peaked and skewed towards the left.
Lastly, in the box plot we can actually see the one outlier.
If we are dealing with outliers, we also check the z-scores. When the z-score is above or
below 1,96/-1,96 then it is a suspicious outlier which is the case for group 1 seeing the
picture below.
> Take into account normally 5% is within this suspicious range, so take 5% of your
observations and see if this really differs from the amount of outliers you have gathered
based on boxplot analysis and z-scores
, - Homogeneity of variance = variance of dependent variable is equal in each
population
a. Levene test
b. Even groups > no problem
c. Largest variance in largest group > overestimation of error variance >
significance is actually lower
d. Largest variance in smallest group > underestimation of error variance >
significance is actually higher
We see in the above example that the largest standard deviation is 2* smallest standard
deviation = Levene test significant so violation of homogeneity.
However, equality of group sizes = valid use of ANOVA > probably Welch test will
therefore show the same results.
, Bonferroni > how many divided by?
Orthogonal
Orthogonal = when you * the coefficients of two contrasts and then add all of them up = 0, if
not > not orthogonal > DELETE NON SIGNIFICANT INTERACTION EFFECT
In this example all contrasts are orthogonal e.g. 1*1 + -1*-1 + 1*-1 + -1*1 = 0
Example of non orthogonal design > see above
> look at type III sum of squares for their own explained variance, because the terms
partially explain the same variance when not orthogonal > see example below
