A statistical method for testing whether two or more A statistical method for testing whether two or more dependent variable means are equal (i.e., the probability dependent variable means are equal (i.e., the probability that any differences in means across several groups are that any differences in means across several groups are due solely to sampling error).due solely to sampling error).

Variables in ANOVA (Analysis of Variance):Variables in ANOVA (Analysis of Variance): Dependent variable is metric.Dependent variable is metric. Independent variable(s) is nominal with two or more Independent variable(s) is nominal with two or more

levels – also called treatment, manipulation, or factor.levels – also called treatment, manipulation, or factor.

One-way ANOVA: only one independent variable with two One-way ANOVA: only one independent variable with two or more levels.or more levels.

Two-way ANOVA: two independent variables each with two Two-way ANOVA: two independent variables each with two or more levels.or more levels.

With ANOVA, a single metric dependent variable is tested With ANOVA, a single metric dependent variable is tested as the outcome of a treatment or manipulation.as the outcome of a treatment or manipulation.

With MANOVA (Multiple Analysis of Variance), two or more With MANOVA (Multiple Analysis of Variance), two or more metric dependent variables are tested as the outcome of a metric dependent variables are tested as the outcome of a treatment(s).treatment(s).

HH00: The means for all groups are the : The means for all groups are the same (equal).same (equal).

HHaa: The means are different for at : The means are different for at least one pair of groups. least one pair of groups.

HH00: : 11 = = 22 = = ……….………. = = kk

HHaa: : 11 22 ……….………. kk

The F-statistic assesses whether you The F-statistic assesses whether you can conclude that statistical differences can conclude that statistical differences are present somewhere between the are present somewhere between the group means.group means.

But to identify where the differences But to identify where the differences are you must use follow-up tests called are you must use follow-up tests called “multiple comparison tests”. Many “multiple comparison tests”. Many multiple comparison tests are available multiple comparison tests are available in SPSS.in SPSS.

Scheffe recommended

Games-Howell recommended

Samples are independent. Samples are independent. Dependent variable is normally distributed for Dependent variable is normally distributed for

each of the samples – with larger sample sizes ( > each of the samples – with larger sample sizes ( > 20/group) not a serious problem should this be 20/group) not a serious problem should this be violated somewhat.violated somewhat.

Whether the sample sizes for the groups are very Whether the sample sizes for the groups are very different (ratio of 1.5 or higher may be a different (ratio of 1.5 or higher may be a problem).problem).

The variances for the different populations from The variances for the different populations from which the samples are drawn are equal – which the samples are drawn are equal – possibly a problem if they are not equal or at least possibly a problem if they are not equal or at least comparable. comparable.

In general ANOVA is a fairly robust procedure.In general ANOVA is a fairly robust procedure.

Application: One-way ANOVA

Variable Description Variable Type

Restaurant PerceptionsX1 Excellent Food Quality MetricX2 Attractive Interior MetricX3 Generous Portions MetricX4 Excellent Food Taste MetricX5 Good Value for the Money MetricX6 Friendly Employees MetricX7 Appears Clean & Neat MetricX8 Fun Place to Go MetricX9 Wide Variety of menu Items MetricX10 Reasonable Prices MetricX11 Courteous Employees MetricX12 Competent Employees MetricSelection Factor RankingsX13 Food Quality NonmetricX14 Atmosphere NonmetricX15 Prices NonmetricX16 Employees NonmetricRelationship VariablesX17 Satisfaction MetricX18 Likely to Return in Future MetricX19 Recommend to Friend MetricX20 Frequency of Patronage NonmetricX21 Length of Time a Customer NonmetricClassification VariablesX22 Gender NonmetricX23 Age NonmetricX24 Income NonmetricX25 Competitor NonmetricX26 Which AD Viewed (#1, 2 or 3) NonmetricX27 AD Rating MetricX28 Respondents that Viewed Ads Nonmetric

Description of Customer Survey VariablesDescription of Customer Survey VariablesVS.VS.

Dependent variable is: XDependent variable is: X2727 – AD Rating – AD Rating Independent variable is XIndependent variable is X2626 – Which AD – Which AD

Viewed (e.g., # 1, 2 or 3):Viewed (e.g., # 1, 2 or 3): ‘ ‘1’ – AD #11’ – AD #1 ‘ ‘2’ – AD #2 2’ – AD #2 ‘ ‘3’ – AD #33’ – AD #3

Research question is: Research question is: Are there differences in the mean Are there differences in the mean

ratings of the ADS based on which AD ratings of the ADS based on which AD was viewed?was viewed?

Phil Samouel asked the researcher to test the effectiveness of three different ads. If the mean ratings of the ads are statistically different he would like to select the highest rated ad and run an advertising campaign for his restaurant.

Not all of the 200 customer respondents agreed to look at and evaluate the ads. To identify the respondents who viewed the ads we go to the Data pull down menu and click on “Select Cases”, then on “If condition satisfied,” then on If. Next highlight variable X28 and click on the Arrow box to move it to the box. Now click on the equal sign (=) below and then one (1) to select only the respondents who viewed the ADS (a zero = did not agree to view ads). Finally click on Continue and then OK.

The metric dependent variable for these hypotheses is X27 — AD Rating and the

nonmetric independent variable is X26 — AD Viewed (# 1, 2 or 3). The click-through

sequence to run the one-way ANOVA is: ANALYZE GENERAL LINEAR MODEL UNIVARIATE. Click on X 27 — AD Rating to highlight it and then on the arrow box to

move it into the Dependent Variable box. Click on X 26 — Which AD Viewed to highlight

it and then on the arrow box to move it to the box labelled “Fixed Factors.” Click on the Post Hoc box and highlight X26 in the Factor(s) box and then click on the Arrow box to

move this variable to the box for Post Hoc Tests. Now look to the lower left side of the screen and click on Scheffe test and Games-Howell and then Continue. Now go to the Options box and click on Descriptive statistics and Homogeneity Tests (Levene test of equal variances) and then Continue, and then click on the Plots box and highlight X26 and move it to the Horizontal Axis box and then under Plots below click Add. Finally, click on Continue and then OK to execute the program.

There is not a significant difference in the variances

of the three groups.

Initial Considerations Initial Considerations – Descriptives & – Descriptives & Levene’s Test of Equal VariancesLevene’s Test of Equal Variances

There are significant differences between ratings for the ads, but

we are not sure where the difference are based on this

test.

There are significant differences between

ratings for all three ads.

Mean Ratings of Ads:

1. Ad #1 = 39.79

2. Ad #2 = 68.03

3. Ad #3 = 51.50

Two-way ANOVA

Examines the effect (if any) of two or more Examines the effect (if any) of two or more nonmetric independent variables on a single nonmetric independent variables on a single metric dependent variable.metric dependent variable.

Total variation is examined for: Total variation is examined for: Variation due to each of the independent Variation due to each of the independent

variables (main effects).variables (main effects). Variation due to the interaction of the Variation due to the interaction of the

independent variables – that is their possible independent variables – that is their possible combined effect on the dependent variable combined effect on the dependent variable beyond the separate influence of each beyond the separate influence of each (interaction effect).(interaction effect).

Variation that remains unexplained (error).Variation that remains unexplained (error).

Three hypotheses are tested Three hypotheses are tested simultaneously:simultaneously:

1.1. The effect of independent variable #1 The effect of independent variable #1 on the dependent variable (main on the dependent variable (main effect).effect).

2.2. The effect of independent variable #2 The effect of independent variable #2 on the dependent variable (main on the dependent variable (main effect).effect).

3.3. The combined (joint) effect of The combined (joint) effect of independent variables #1 and #2 on independent variables #1 and #2 on the dependent variable (interaction the dependent variable (interaction effect).effect).

Main EffectMain Effect = the impact any single = the impact any single experimental variable has on a response experimental variable has on a response (dependent) variable.(dependent) variable.

Interaction EffectInteraction Effect = the combined impact of = the combined impact of multiple independent variables on a response multiple independent variables on a response variable; i.e., is the difference in the mean variable; i.e., is the difference in the mean ratings of the ads (response variable) the same ratings of the ads (response variable) the same when we compare males and females?when we compare males and females?

Blocking VariableBlocking Variable = a grouping variable the = a grouping variable the researcher doesn’t manipulate or control in any researcher doesn’t manipulate or control in any way, such as gender.way, such as gender.

Phil Samouel asked the researcher to test three different ads for their effectiveness. If the ratings of the ads are statistically different he would like to use that information to attract more customers. He also would like to know how various demographic characteristics are related to ad ratings. In this case, we use gender, which is referred to as a blocking variable. The null hypotheses are:

(1) No differences in ad ratings based on which ad was viewed; (2) No differences in ad ratings based on gender; (3) No differences in ad ratings based on the combined effects of

which ad viewed and gender.

The metric dependent variable for these hypotheses is X27 — AD Rating and the nonmetric independent variables are X26 — AD Viewed (# 1, 2 or 3) and X22 — Gender.

Recall that not all of the 200 customer respondents agreed to look at and evaluate the ads. To identify the respondents who viewed the ads we go to the Data pull down menu and click on “Select Cases”, then “If condition satisfied,” then on If. Next highlight variable X28 and click on the Arrow box to move it to the box. Now click on the equal sign (=) below and then one (1) to select only the respondents who viewed the ADS (a zero = did not agree to view ads). Finally click on Continue and then OK.

The click through sequence is: ANALYZE GENERAL LINEAR MODEL UNIVARIATE. Highlight the dependent variable X27 — AD Rating by clicking on it and move it to the Dependent variable box. Next, highlight X26 — AD Viewed and X22 — Gender, and move them to the box labelled “Fixed Factors.” Now click on the Post Hoc box and highlight X26 in the Factor(s) box and then click on the Arrow box to move this variable to the box for Post Hoc Tests. We do not move X22 because it has only two groups and not three. Look to the lower left side of the screen and click on Scheffe test and then Continue. Now go to the Options box and click on Descriptive statistics and then Continue, and then click on the Plots box and highlight X26 and move it to the Horizontal Axis box and then click the Add button above the Plots box below. Finally, click on Continue and then OK to execute the program.

Mean ratings of ads by which ad viewed

and gender.

Sample sizes for each of the groups.

• AD Rating main effect significant (X26).

• Gender main effect not significant (X22).

• Interaction effect significant (X26 * X22).

If the interaction effect is not significant, the

main effects of the treatments are

independent and can be interpreted directly.

If the interaction effect is significant,

then the type of interaction must be

determined.

The significant interaction and nonsignificant main effect for X22

raises a red flag.

All comparisons significantly different.

The three ads are rated differently, with ad #1 rated lowest at 39.79, #3 somewhat higher at 51.50, and #2 the highest at 68.03.

There is a difference in ratings by gender across all three ads, with female ratings overall slightly more favorable (55.55 vs. 54.56). But remember overall there was not a statistically significant difference.

There is a significant difference between

AD Ratings by males and females for ads

#1 and #3, but not for ad #2.

Note: do not be fooled by the

slope of the line – the mean rating for males is 68.5 and for females

is 67.8.

1. When should ANOVA be used?1. When should ANOVA be used?

2. What is the difference between one-way and2. What is the difference between one-way and two-way ANOVA? two-way ANOVA?

3. What are “multiple comparison tests” and3. What are “multiple comparison tests” and why are they used? why are they used?

4. What is the difference between a main effect 4. What is the difference between a main effect and an interaction effect? and an interaction effect?