Chapter 12: Introduction to Analysis of Variance (ANOVA)

True or False? (B) If the null hypothesis is true, the F-ratio for ANOVA is expected (on average) to have a value 0

(False) [If the null hypothesis is true, the F-ratio will have a value near 1]

True or False? (A) Post Hoc tests are needed if the decision from an analysis of variance is "fail to reject the null hypothesis".

(False) [Post Hoc tests are needed only you reject H0 (indicating at least one mean difference is significant)]

True or False? ANOVA allows researchers to compare several treatment conditions without conducting several hypothesis tests.

(True) [Several conditions can be compared in one test]

True or False? A report shows ANOVA results: F(2, 27) = 5.36, p − value < 0.05. You can conclude that the study used a total of 30 participants.

(True) [dfbetween = k − 1 = 2; dfwithin = N − k = 27; dftotal = dfbetween + dfwithin = (k − 1) + (N − k) = N − 1 = 29 → N = 30]

Post Hoc Tests

(i) A method for testing which populations have different means after the one-way ANOVA rejected the null hypothesis. (ii) A significant F-ratio (i.e., when H0 is rejected) indicates that at least one difference in means is statistically significant. (iii) Post hoc tests are follow-up tests done to determine exactly which mean differences are significant, and which are not.

One-way ANOVA Assumptions:

(i) All populations are normally distributed. (ii) The population variances are equal. (iii) The observations are independent - that is, the occurrence of any one individual value does not affect the probability that any other observation will occur. (iv) The data are interval or ratio level.

Measuring Effect Size

(i) Compute percentage of variance accounted for by the treatment conditions. (ii) In published reports of ANOVA, effect size is usually called η 2 ("eta squared"). (iii) Effect Size: η^2 = SSbetween / SStotal

Experimental Study:

(i) Definition: An experiment is a study in which the experimenter manipulates attributes of the study participants and observes the consequences. Designed experiments are conducted to demonstrate a cause-and-effect relationship between one or more explanatory factor (or predictors) and a response variable. (ii) Example: Vitamin C and Cold Prevention The effects of vitamin C on the prevention of colds in 868 children was conducted in 1976. Of the 868 children studied, half were randomly selected for the experimental group. Children in this group received a 1,000-mg tablet of vitamin C daily for the test period. The remaining children, who made up the control group, received a placebo - an identical tablet containing no vitamin C - also on a daily basis. The results showed that the average number of colds per child was .38 for children receiving vitamin C, while the average for children receiving the placebo was .37 leading to non-significant statistical difference

Type I Error and Multiple Hypothesis Tests:

(i) Experiments often require multiple hypothesis tests—each with Type I error (α). (ii) Type I error for a set of tests accumulates testwise α. (iii) ANOVA evaluates all mean differences simultaneously with one test—regardless of the number of means—and thereby avoids the problem of inflated experimentwise α.

Logic of Hypothesis Test:

(i) If the null hypothesis is true, the populations have identical distributions - sample means for random samples from each population should be close in value. (ii) The null hypothesis should be rejected only if the sample means are substantially different (some pairs may be the same).

Analysis of Variance

(i) Used to evaluate mean differences between two or more treatments. (ii) Uses sample data as basis for drawing general conclusions about populations

Experimental Unit:

An experimental unit is the object on which a measurement (or measurements) is taken.

Factor:

An explanatory or independent variable whose effect on the response variable is to studied and whose values are controlled and varied by the experimenter

Which combination of factors is most likely to produce a large value for the F-ratio? (A) Large mean differences and large sample variances. (B) Large mean differences and small sample variances. (C) Small mean difference and large sample variances. (D) Small mean differences and small sample variances.

B

Observational Study:

Definition: A statistical study is observational in which the researcher simply observes the subjects without interfering and the data are collected without any particular design. (ii) Example: Consider someone on the busy street of a New York neighborhood asking random people that pass by how many pets they have, then taking this data and using it to decide if there should be more pet food stores in that area. This is an observational study because the researcher is simply observing the answers of the survey without influencing the outcome in any way.

Between Sample Variation (SSbetween):

Dispersion among the factor sample means.

F-ratio Interpretation:

If H0 is true, size of treatment effect is near zero, hence F=1 If HA is true, size of treatment effect is more than zero, hence F= >>1

Level:

Intensity setting or different values of a factor.

n

Number of observations in each treatment/level. (a) If the number of observations in each treatment is different (n1, n2, . . . , nk ), the design is unbalanced. (b) If the number of observations in each treatment is same (i.e., n1 = n2 = . . . = nk = n), the design is balanced

k

Number of treatments

Treatment:

Specific combination of factor levels.

Total Variation (SS-total):

The aggregate dispersion of the individual data values across the various factor levels.

Within Sample Variation (SSwithin):

The dispersion that exists among the data values within a particular factor level.

Response:

The values of the outcome variable being measured by the experimenter.

T

Treatment Total