Business 205

Review Analysis of Variance (ANOVAs)

Preview 2 factor ANOVAs Reporting Excel Data Analysis ToolPak 2 Independent Sample T-tests ANOVAs 2-Factor ANOVAs

1-Factor ANOVAs Looked at different levels of ONE IV. Temperature: 30, 40, 50 degrees Manager Interaction: Low, Medium, High Products: Pepsi, Coke Compared the different levels of the 1 IV to each other to see if things were significant.

Scenario You are a manager and want to study factors that affect a worker’s performance. Some workers have mentioned that when they are hot, they can’t work as hard while other workers have mentioned that sometimes they have a difficult time seeing because there isn’t enough light so they aren’t as productive as they should be. What are the IVs and DV?

2-Factor ANOVAs You have more than 1 IV You are looking at different levels within the different IVs Lighting: Low (60 watt) vs. Bright (125 watt) Temperature: 70 degrees vs. 80 degrees 2 x 2 Factorial Design

Factorial Designs 2 x 2 2 light (60 watt/125 watt) x 2 temperature (70 degrees/80 degrees) 3 x 2 3 light (60 watt/80 watt/100 watt) x 2 temperature (70 degrees/80 degrees) 3 x 3 3 light (60 watt/80 watt/100 watt) x 3 temperature (60 degrees/70 degrees/80 degrees)

2-Factor ANOVA Main Effect What effect each of the factors has on the DV Main effect for temperature on work performance. Main effect for lighting on work performance Interactions The mean differences between treatment conditions are different than what is predicted from the overall main effects Temperature x Lighting

2-Factor ANOVA Hypotheses You can have a hypothesis for each IV Example: IVs: Temperature, Lighting H1: Temperature will affect work performance H2: Lighting will affect work performance You can have a hypothesis for each interaction H3: There will be an interaction between temperature and lighting that will affect work performance.

Reporting 2-Factor ANOVAs in table form SourceSSdfMSF Between treatment2205 Factor A (lighting) Factor B (temp) A x B interaction Within treatment Total56033

Stating Results for 2-Factor ANOVAs SourceSSdfMSF Between treatment2205 Factor A (lighting) Factor B (temp) A x B interaction Within treatment Total56033 You now have a conclusion for EACH of the hypotheses which means in a 2 factor ANOVA, you have 3 critical F values, 3 graphs with critical regions, and 3 conclusions: 1 for the 1 st IV, 1 for the 2 nd IV, and 1 for the interaction.

2-Factor ANOVA Assumptions 1. The observations within each sample are independent 2. The populations from which the samples are selected are normal 3. The populations from which the samples are selected have equal variances

Excel Data Analysis ToolPak 2 Independent Sample T-tests Single Factor ANOVAs 2 Factor ANOVAs

Formulas for 2 Factor ANOVAs You will NOT be asked to do a 2 Factor ANOVA by hand on the exam. You will need to know general information about a 2 Factor ANOVA. The following slides are for your edification only.

Example You are a manager and want to study factors that affect a worker’s performance. Some workers have mentioned that when they are hot, they can’t work as hard while other workers have mentioned that sometimes they have a difficult time seeing because there isn’t enough light so they aren’t as productive as they should be.

Defining our levels within each IV Lighting: Low (60 watt) Medium (75 watt) High (100 watt) Temperature Hot (80 degrees) Cold (60 degrees) What type of design is this? ____ X _____

Structure of 2-Factor ANOVA Total Variability Between-treatments Variability Within-treatments Variability Factor A Variability Factor B Variability Interaction Variability Stage 1 Stage 2

Components of an ANOVA SymbolDefinition knumber of treatment conditions nnumber in each treatment condition Ntotal number in the study (across all conditions) Tsum of each individual score per treatment SSsum of squares (X – Mean) 2 for each treatment Ggrand total; sums of all scores in an experiment ∑X 2 each individual score squared then summed for each treatment

Formulas k = ∑ all treatments N = ∑ n for all treatments n = number of scores in each INDIVIDUAL treatment T = ∑ X (all scores in each INDIVIDUAL treatment) SS = ∑ (X-M) 2 for each treatment M = mean for each treatment G = ∑ T ∑ (X 2 ) = sum of all individual scores squared in all treatments

Data LowMediumHigh Hot 5 3 T = 25 8 SS = T = 45 6 SS = T = 20 3 SS = 20 3 Cold T = 5 0 SS = T = 5 5 SS = T = 20 5 SS = 28 5 Lighting (B) Temp (A) T Low = 30 T Med = 50 T high = 40 T Hot = 90 T cold = 30 N = 30; G = 120; ∑X 2 = 820

Stage 1 Calculations Run a “normal” ANOVA to find: df between, df within and df total SS between, SS within and SS total MS within

Degrees Freedom df between = number of cells -1 6 – 1 = 5 df within = ∑df for all treatments = 24 df total = df between + df within = 29

Sums of Squares Formulas 820 – ((120 2 )/30) = = 120

Mean Squares 120/4 = 5.00

Stage 2 Analysis Compute for Factor A df between A SS between A MS between A F A Compute for Factor B df between B SS between B MS between B F B Compute the Interaction of Factor A x Factor B

Factor A df between A = number of levels of A -1 2 – 1 = 1 =

Factor B df between B = number of levels of B -1 3 – 1 = 2 =

A x B Interaction df AxB = df between treatments - df A - df B 5 – 1 – 2 = – 120 – 20 = 80

Finding the F-ratios

Significant Values Consult the F-distribution table for EACH F-test result using proper dfs in each case.