ANOVA Hypothesis Testing-Statistic Homework
mehett
lecture_anova.pptxANOVA
Instead of looking at the difference between population means, ANOVA (analysis of variance) calculates the variance between population means
ANOVA
Sample pop. 1
Sample pop. 2
Sample pop. 3
H0: μ1 = μ2 = μ3
ANOVA
Calculating ANOVA is different than calculating t-tests
How much sampling error do we expect under H0? (i.e., how much should our sample(s) vary just by chance?)
How do we calculate the equivalent of the standard error?
t-tests
Observed difference in sample means
Expected difference in pop. means
ANOVA
Observed variance in sample means
Expected variance in pop. means
ANOVA
Rock
Country
Classical
Mclassical
16.4
Mrock
6.0
Mcountry
10.8
Moverall 11.07
Under H0, all of these scores come from the same H0 distribution with M = 11.07. Based on the spread of each group, does that seem likely?
Under H0, all of these scores come from the same H0 distribution with M = 11.07. Based on the spread of each set, does that seem likely?
3
Score
ANOVA
Rock
Country
Classical
Mclassical
16.4
Mrock
6.0
Mcountry
10.8
Under H0, all of these scores come from the same H0 distribution with M = 11.07. Based on the spread of each group, does that seem likely?
Moverall 11.07
Under H0, all of these scores come from the same H0 distribution with M = 11.07. Based on the spread of each set, does that seem likely?
4
Score
ANOVA
Rock
Country
Classical
Mclassical
16.4
Mrock
6.0
Mcountry
10.8
Under H0, all of these scores come from the same H0 distribution with M = 11.07. Based on the spread of each group, does that seem likely?
Moverall 11.07
Under H0, all of these scores come from the same H0 distribution with M = 11.07. Based on the spread of each set, does that seem likely?
5
Score
ANOVA
The amount of variation within each group contributes to the expected variance in population means
Steps of hypothesis testing
Select the appropriate test
ANOVA used for 3+ means
State your research hypothesis and your null hypothesis
State them in English, then in math
Describe the NULL distribution
Variance between means expected by chance
Compute your test statistic
Compute actual variance between means
Then compute test statistic
Determine your critical value
Compare the test stat to the critical value(s) and make your decision
1. Select Test
We’re testing 3+ means, so…
ANOVA
2. State your hypotheses
H0: There are no differences in the means
H1: There are differences between the means
(You should add in the content appropriate to the question)
H1: At least one μi ≠ μj
H0: μ1 = μ2 = μ3
3. Describe the Null
For ANOVA: What is the variance between means expected by chance?
We compute this based on the within-group variance
Roughly, it’s like computing the variances within each group and averaging them together
(But not quite)
So let’s see how we actually compute it…
3. Describe the null
The variance within is called the Mean Square Within or MSWithin
Numerator is Sum of Squares Within or SSWithin
Denominator is Degrees of Freedom Within or dfWithin
Looks just like a variance!
Looks just like a sum of squares!
4A. Compute actual variance between means
What is the actual variance between your 3+ means?
This is the between group variance
Roughly, we compute this by treating the group means as raw scores…
And then literally calculate the variance of these “raw scores”
4A. Compute actual variance between means
The variance between is called the Mean Square Between or MSbetween
Numerator is Sum of Squares between or SSbetween
Denominator is Degrees of Freedom Between or dfbetween
Looks just like a variance!
Looks almost like a sum of squares!
Intermission: Check your work
You can calculate SStotal independently and make sure your answers match!
4B. Compute test statistic
For ANOVA, it is an F-test
Variance between
Variance within
Variance (mean of squared deviations)
dftotal = dfbetween + dfwithin
SStotal = SSbetween + SSwithin
Sources of estimates of population variability
How to report an F-test
The ANOVA source table
| Source | SS | df | MS | F |
| Between | SSbetween | dfbetween | MSbetween | F |
| Within | SSwithin | dfwithin | MSwithin | |
| Total | SStotal | dftotal |
5. Determine critical values
F-tests are always one-tailed
Variances can never be less than zero
To look it up, you’ll need
α, dfwithin, dfwithin, and Table D
6. Make decision
If F > critical value, reject null H0
If F < critical value, retain H0
The F-distribution
If H0 is true, all samples come from the same population
Little variance between samples compared to variance within samples
If H0 is not true, samples come from different populations
More variance between samples than variance within samples
Variance between
Variance within
F =
F ≤ 1
F > 1
The F-distribution
Unlike t- and z-distributions, the F-distribution is asymmetrical
Variances are always positive, so the ratio of variances is always positive
Therefore the F-distribution is…
positively skewed
Variance between
Variance within
F =
The F-distribution
Notice that the shape of the distribution makes all tests one-tailed
ANOVAs cannot be used to test directional hypotheses
The F-distribution
Like t-distributions, there is a family of F-distributions specified by:
dfbetween (# of groups – 1)
dfwithin (# of scores in each group – 1)
The more datapoints, the better the estimates of the population variance
Distributions with lots of data have taller peaks and shorter tails
The F-distribution
dfbetween = 4, dfwithin = 4
dfbetween = 10, dfwithin = 4
dfbetween = 10, dfwithin = 10
dfbetween = 4, dfwithin = 10
For your homework, report:
Steps of hypothesis testing
ALL the equations on the next slide
The full ANOVA source table
The critical value
A comparison of the F-stat to the critical value
A description of what you found
See final slide for example
Note: Mgroup changes depending on which group a raw score comes from. Ngroup changes depending on which group the Mgroup comes from
Steps of hypothesis testing
Select the appropriate test
ANOVA used for 3+ means
State your research hypothesis and your null hypothesis
State them in English, then in math
Describe the NULL distribution
Compute MS, SS, and df – all the within versions
Compute your test statistic
Compute MS, SS, and df between; check your work using SS and df total
Compute F statistic
Determine your critical value
Compare the test stat to the critical value(s) and make your decision
How to report an F-test
The ANOVA source table
| Source | SS | df | MS | F |
| Between | SSbetween | dfbetween | MSbetween | F |
| Within | SSwithin | dfwithin | MSwithin | |
| Total | SStotal | dftotal |
For every problem, report this table. Replace all SS, df, MS and F in the light pink/purple cells with the actual numbers.
Describing your result
“This hypothesis test shows that we should reject the null hypothesis and instead conclude that SUNY campuses differ in intelligence. This conclusion is statistically significant with p < .05.”
“This hypothesis test shows that we should retain the null hypothesis and conclude that SUNY campuses do not differ in intelligence. The test was not statistically significant with p > .05.”
Replace underlined segment with content from the particular problem
