ANOVA:Full FactorialPrepared By:106 Pranav Jain151 Aayush Agarwal

167 Surbhi Koshal506 Harsh Gupta

Terminology Response variable

Measured output value Factors

Input variables that can be changed Levels/Treatments

Specific values of factors (inputs) Replication

Completely re-run experiment with same input levels Used to determine impact of measurement error

Interaction Effect of one input factor depends on level of another input

factor

ANOVA Statistical technique used to investigate and model relationship between a response variable and one or more independent variables

Each independent variable is called as factor Each factor has two or more levels

Hypothesis H0: Population means of each level are equal H1: At least one of the level means are not all equal

Why use ANOVA instead of t test? ANOVA and two sample t test give identical result for mean with one factor and two levels

But real world modeling require more than one factor

Thus, ANOVA can simultaneously test several factors each with several levels

Steps in design of experiments1. Identify factors of interest and response

variable2. Determine appropriate levels of each

explanatory variable3. Determine design structure4. Randomize the order in which each set

conditions is run and collect the result data5. Organise the results to draw appropriate

conclusions

Factorial Design : Full factorial design A full factorial design of experiments consists of the following: Vary one factor at a time Perform experiments for all levels of all factors Hence perform a large number of experiments that are

needed! All interactions are captured

Consider a simple design for the following case: Let the number of factors = k Let the number of levels for the ith factor = ni

The total number of experiments (N) that need to be performed is

K

iinN

1

Factorial Essentials Notation System E.g. 2x3, 2 IV, one with 2 level, one with 3 and 6

total conditions• Factorial Matrix

2x2 presentation rate 2-sec/word 4-sec/word

type of Imagerytraining Rote

* Dependent variable: Words remembered by viewer

Main Effects Overall effect of an Independent Variable (IV) on a dependent variable

Main Effect Calculation Main effect of training effect = 20-15 = 5 Imagery produces better results than rote by 5 words

Main effect of presentation rate = 14.5-20.5 = -6 2-sec/word produces a worse result than 4-sec/word by -6

Interactions Teaching Style

Lab Lectures

Science Humanity

*No main effect is present in the above case“the size or direction of the simple main effect on DV of IV1 changing at various levels of IV2”

80 70

70 80

InteractionsMajor at Lab 80 70 = -10Major at Lects 70 80 = 10 Major at Science 80 70 = -10 Major at Humanity 70 80 = 10

Science Humanity65

70

75

80

85

Interaction Effects

Lab Lectures Mean

Interaction effect Interaction effect occurs when one factor effects the results differently depending upon the second factor

No interaction between exercise and drugs

0

2

4

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A B C DExercise

Inte

nsity

Drug 1 Drug 2 Drug 3 Drug 4

0.0

5.0

10.0

15.0

20.0

25.0

30.0

A B C D

Exercise

Inte

nsity

Drug 1 Drug 2 Drug 3 Drug 4

Subtle Effect

0.0

1.0

2.0

3.0

4.0

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7.0

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A B C DExercise

Inte

nsity

Drug 1 Drug 2 Drug 3 Drug 4

Dramatic Interaction

Experiment

Experiment Information on the earnings of men and women for different occupations was collected. Suppose that the aim of the experiment is to investigate whether there were any differences between the weekly salaries ($) of men and women employed as computer programmers, contractors and doctors.

A sample of 10 men and 10 women was selected from each of the three occupations, and the weekly salary for each individual in the sample was recorded.

Considering 0.05 level of significance, we need to test for any significant effect due to occupation, gender and interaction.

Terminology Response variable

Weekly Income ($) Factors

A) Gender B) Occupation Levels

A) Male | Female B) Comp Programmer | Contractor | Doctor

Replication 60/(2x3) = 10 replications

Interaction Dependency of effect of gender on different occupations or Dependency of effect of occupations on different gender

Hypothesis Gender H0: Average salary for males and females are same. H1: Average salary for males and females are different. Occupation H0: Average salary for doctors, contractors and

computer programmers is same. H1: Average salary for doctors, contractors and

computer programmers is different. Interaction H0: There is no interaction effect taking place. H1: There is some interaction effect taking place.

Data Set

Between-Subjects Factors  N

OccupationComp Programmer 20Contractor 20Doctor 20

Gender Female 30Male 30

Microsoft Excel Worksheet

Data SummaryDescriptive Statistics

Dependent Variable: Weekly SalaryOccupation Gender Mean Std.

Deviation N

Comp Programmer

Female 741.30 78.942 10Male 796.00 86.340 10Total 768.65 85.267 20

ContractorFemale 634.70 109.283 10Male 979.40 104.198 10Total 807.05 205.104 20

DoctorFemale 930.90 100.507 10Male 1046.90 91.001 10Total 988.90 110.674 20

TotalFemale 768.97 155.878 30Male 940.77 140.991 30Total 854.87 170.932 60

AnalysisTests of Between-Subjects Effects

Dependent Variable: Weekly SalarySource Type III Sum

of Squaresdf Mean

SquareF Sig. Partial Eta

Squared

Corrected Model 1230024.533a 5 246004.907 26.901 .000 .714

Intercept 43847821.067 1 43847821.0

674794.80

5 .000 .989

Occupation 553693.633 2 276846.817 30.273 .000 .529Gender 442728.600 1 442728.600 48.413 .000 .473Occupation * Gender 233602.300 2 116801.150 12.772 .000 .321

Error 493822.400 54 9144.859      Total 45571668.0

00 60        

Corrected Total 1723846.933 59        

a. R Squared = .714 (Adjusted R Squared = .687)

Profile Plots: Interaction EffectThe slopes for each level of occupation are different from each other, hinting at existence of interaction effects.Crossing of 2 lines (Programmer and Contractor) indicate significant effect on wages due to gender in the occupations.This might be attributed to the job requirements of the occupation

Profile Plots: Interaction EffectThe slopes for each level of occupation are different from each other, hinting at existence of interaction effects.Inverse movement of 2 lines (Programmer and Contractor) indicate significant effect on wages due to gender in the occupations.This might be attributed to the job requirements of the occupation

Conclusion Gender Reject H0 -> Average salary for males and females are different Occupation Reject H0 -> Average salaries for doctors, contractors and

computer programmers are different Interaction Reject H0 -> There is some interaction effect taking place between

gender and occupation

Thank You