One Way Anova

PSYC 2317 Mark W. Tengler, M.S.

Assignment #14 One Way Anova F Test

14.1 Describe the similarities and differences between an F-ratio and a t statistic.

14.2 Explain why you should use ANOVA instead of several t tests to evaluate mean differences (i.e. differences between treatment groups) when an experiment consists of three or more treatment conditions.

14.3 A common science-fair project involves testing the effects of music on the growth of plants. For one of these projects, a sample of 24 newly sprouted bean plants is obtained. These plants are randomly assigned to four treatments, with n = 6 in each group. The four conditions are rock, heavy metal, country, and classical music. The dependent variable is the height of each plant after 2 weeks. The data from this experiment were examined using an ANOVA, and the results are summarized in the following table. Fill in all of the missing values. (Hint: start with the df column using the formula sheet part III.)

Source SS df MS or s2 F___ Between treatments 60 ____ ____ ____ Within treatments ____ ____ 2 Total ____ ____

14.4 First-born children tend to develop language skills faster than their younger siblings. One possible explanation for this phenomenon is that first-borns have undivided attention from their parents. If this explanation is correct, then it is also reasonable that twins should show slower language development than single children and that triplets should be even slower. Davis (1937) found exactly this result. The following hypothetical data demonstrate the relationship. The dependent variable is a measure of language skill at age 3 for each child. Do the data indicate any significant differences? Test with α

Single Child Twin Triplet 8 4 4 7 6 4 10 7 7 6 4 2 9 9 3

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Three or More Independent Groups with One Independent Variable Design One Way Analysis of Variance (ANOVA) - F ratio test

I. Assumptions for F-test A. Populations

1. Three or more populations from which the samples are selected are normal 2. Three or more populations must have equal variances (i.e. are the same

populations) 3. only the mean of the scores is affected by the independent variable, not the

variance B. Three or more random samples (each tested once) derived by:

1. Three or more different samples from same population (looking to see if populational differences due to treatment or chance)

C. Data values 1. Sample values known (mean, standard deviation) 2. Populational values (mean, standard deviation) not known

II. Diagramming your research (shows the whole logic and process of hypothesis testing)

a. Draw a picture of your research design (see diagramming your research handout). b. There are always two explanations (i.e. hypotheses) of your research results, the

wording of which depends on whether the research question is directional (one-tailed) or non-directional (two-tailed). State them as logical opposites.

c. For statistical testing, ignore the alternative hypothesis and focus on the null hypothesis, since the null hypothesis claims that the research results happened by chance through sampling error.

d. Assuming that the null is true (i.e. that the research results occurred by chance through sampling error) allows one to do a probability calculation (i.e. all statistical tests are nothing more than calculating the probability of getting your research results by chance through sampling error).

e. Observe that there are two outcomes which may occur from the results of the probability calculation (high or low probability of getting your research results by chance, depending on the alpha (α) level).

f. Each outcome will lead to a decision about the null hypothesis, whether the null is probably true (i.e. we then accept the null to be true) or probably not true (i.e. we then reject the null as false).

III. Hypotheses

A. One-tailed (all F ratio’s are one-tailed) 1. Alternative hypothesis (H1): The independent variable does make a difference in

performance between the groups. 2. Null hypothesis (H0): The independent variable does not make a difference in

performance between the groups other than by sampling error.

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IV. Determine critical regions (i.e. critical F ratio value between high & low probability) using table

B.4 (pp. A-29ff) A. Significance level (should be given or decided prior to experiment)

1. α or p levels a. α = .05, use lightface type for critical F value b. α = .01, use boldface type for critical F value

B. Degrees of freedom 1. Numerator (between groups)

a. dfBet = k (# of groups) - 1 2. Denominator (within groups, dfW)

a. dfWith = (n1 + n2 + ... + nk) - k (# of groups) C. With degrees of freedom and α level, find the critical F ratio value

V. Calculate F-test statistic (One-way ANOVA)

A. General statistical test formula between groups variance = treatment effect + chance

F ratio = within groups variance chance B. F-test formula for more than two independent groups with one independent variable

F = sBetween 2

sWithin 2

C. Calculations 1. calculating SSBetween: between all groups Sum of the Squares estimate

SSBetween = (∑x1)

2 + (∑x2) 2 + (∑xk)

2 _ (∑x1 + ∑x2 +∑xk) 2

n1 n2 nk n1 + n2 + nk

2. calculating SSWithin: within groups Sum of the Squares estimate

SSWithin = SS1 + SS2 + SSk or s1 2(n1-1) + s2

2(n2-1) + sk 2(nk-1)

where SS = ∑x2 _ (∑x)2 for each treatment group

n D. Make table of variances and calculate appropriate F-ratios

Source

SS

Df

s2 or MS

F ratio Between Groups

SSBetween

dfBetween

sBetween

2 = SSBetween dfBetween

F = sBetween

2 sWithin

2 Within Groups

SSWithin

dfWithin

sWithin

2 = SSWithin dfWithin

Total

Add Between & Within

Add Between & Within

3

VI. Compare calculated F-statistic to critical F-value (from A-29 table) & make decision

A. Reject null and accept alternative or B. Accept null

VII. Reporting the results of a three or more independent groups t test

“The means and standard deviations are presented in Table 1. The analysis of variance revealed

a significant difference, F(dfBetween,dfWithin) = 3.67, p < .05.”

Table 1

Independent Variable Levels

Control

Treatment A

Treatment B

M

5.0

8.0

7.5

SD

1.73

1.86

1.99