Flashcards in Lecture 6 Deck (31):
What are the ANOVA assumptions?
2. Homogeneity of variance
3. Independence of observations
4. DV measured on an interval or ratio scale
5. X (IV) & Y (DV) are linearly related
Explain the normality assumption of ANOVA.
- For any value of x (the IV aka the raw scores) are approximately normally distributed.
- In other words, the raw scores are normally distributed with in each group. Do a frequency distribution of the raw scores for each group.
What is the effect of violation of the normality assumption of ANOVA on type I and type II errors?
Type I error:
- non normality only has a slight effect on type I error.
- even for very skewed, or kurtotic (peakedness) distributions.
e.g. nominal alpha (what we set alpha at = type I errror when all assumptions met) vs actual alpha (type I error if one or more assumptions are violated)
In really non-normal populations, when nominal alpha = .05, actual alpha = .055 or .06. If nominal alpha ~ actual alpha, what do we say?
We say F is robust to violations of the assumptions.
Therefore F is robust with respect to the normality assumption.
What are the reasons that F is robust with respect to the normality assumption?
The sampling distribution of the mean will be normally distributed if:
a) the raw scores are normally distributed in the population.
b) The raw sores in the population are skewed, the sampling distribution of the mean will approach a normal distribution as n increases (n greater than or equal to 30 or so).
Define standard error of the mean:
The standard deviation o the sampling distribution of the mean.
When would you use a non-parametric test? Why?
When the population is very skewed. Because non-parametric tests are distribution free, which means they don't have the normality assumption.
What effect does lack of normality have on power?
- Only a light effect (a few 100ths)
- Lack of normality due to platy kurtosis (flattened distribution) does affect power, especially if n is small.
How does one check for normality?
- Check via frequency distributions
- If big violation of normality with small n --> conduct a non-parametric test --> i.e. distribution doesn't matter.
What are some examples of non-parametric tests?
Describe the Homogeneity (homoscedasticity) of variance assumption
i.e. variance (refers to error variance aka within group variance) is unaffected by the treatment i.e. the IV
error due to chance
variability due to chance
i.e. σ²1 = σ²2 = σ²3 etc
In other words, for every value of x, the variance of y is the same.
Illustration of heteroscedasticity
Scores (y axis) and independent variable (x axis)
Each group's scores grouped together above each group
Under what circumstances is F robust for unequal variances?
If n's are equal or approximately equal.
When is heterogeneity of variance an issue?
Only an issue if:
- n's are sharply unequal and a test shows that the variances are sharply unequal.
What is meant by approximately equal n?
largest n/smallest n < 1.5
What is meant by approximately equal σ²? (variance)
Largest variance/smallest variance > 3
If ratio is greater than 3, we have sharply unequal σ².
If Fmax > 3, then the variances are sharply unequal
When is heterogeneity an issue for type I error? (case 1)
Case 1: If the largest variance is associated with the group with smallest n
F is liberal. i.e. actual alpha is going to be greater than nominal alpha.
i.e. falsely reject H0 too often
Solution: adjust nominal alpha downwards. e.g. .025 --> therefore actual alpha is approximately .05
When is heterogeneity an issue for type I error? (case 2)
Case 2: If largest variance is associated with the group with largest n
F is conservative.
i.e. actual alpha is less than nominal alpha.
So people usually don't make an adjustment.
Explain the independence of observations assumption of ANOVA
- Observations within each group are independent of one another.
- Usually satisfied if unrelated subjects run individually and alone.
- Usually satisfied if subjects run individually and alone.
Why is the independence of observations assumption of ANOVA so important?
Because even small violations have a substantial effect on both alpha and power.
How is dependence measured?
Explain the DV is measured on an interval or ratio scale assumption of ANOVA
Check definitions against actual DV used.
If DV is nominal or ordinal, conduct a different type of statistical test. e.g. Chi square test
Explain the X (IV) & Y (DV) are linearly related assumption of ANOVA
i.e. a subject's score is comprised of 3 parts:
1. general effect (grand mean)
2. an effect that is unique and constant within a given treatment.
3. An effect that is unpredictable (random error & individual differences).
Give the linear model of the fifth assumption for ANOVA
μ + alphaj + eij
where μ = grand mean
alphaj = treatment effect for the jth group
and eij = random error for the ith subject in the jth group
general effect + treatment effect + error
Define an outlier
a data point which is very different from the rest of the data.
Outliers can have a dramatic effect on results
When removing outliers, what must be done?
Must explain why they were removed, this information must be shared.
What causes outliers?
1. Human error (eg data entry)
3. Subjects significantly different from the rest of the sample --> perhaps from a different population.
Therefore need to detect and remove outliers.
How do you detect outliers for small samples?
- The largest possible z score of a data set is bounded by: (n-1)/√n
eg. for n=10, largest possible z score is 2.846, therefore for small samples, scrutinize any data point greater than or equal to z=2.5
How d you detect outliers for large, normally distributed samples?
- Approximately 99% of scores are within three standard deviations of the mean.
Therefore z scores >3 should be scrutinized
Note: If n>100 will get some z scores >3 by chance.
A criteria of z>3 is also reasonable for non-normal distributions, but could extend it to z>4.
What happens when subjects are run after analyses?
Tends to increase variability, therefore decrease probability of finding significance AND if N is really large, tend to get statistical significance no matter what, even if no practical significance.