** FACTORIAL DESIGN**

If two or more independent variables are to be tested through an ANOVA, we use a Factorial Design. The test is similar to Two-way ANOVA. A two way ANOVA can estimate the main effects of independent variable, but the factorial design can estimate the main effects along with the interaction effect of the independent variables. Factorial designs are described using “A B” notation, “A” stands for the number of levels of one independent variable and “B” stands for the number of levels of the second independent variable. There would be three null hypotheses for two main effects and the interaction with ‘all means equal’.

**Type of the Variable**

The dependent variable should be in interval or ratio scale and the independent variables should be in nominal scale.

**CASE ANALYSIS-1**

**PROBLEM**

A study is conducted to find the effects of brand and flavour on sales of ice-creams. The purpose is to estimate:

- The effect of the independent variables ‘brand’ and ‘flavour’ on the dependent variable ‘sales’. (Main effect)
- The combined effect of ‘brand’ and ‘flavour’ on sales. (Interaction effect)

The codes used for the independent variables are as follows.

**Flavour**: Vanilla (Code =1), Chocolate (Code=2), Strawberry (Code=3), Butter scotch (Code=4).

**Brand:** Vadilal (Code =1), Amul (Code =2), Kwallity (Code =3), Milko (Code =4).

There are ** **level of combinations of flavor and brand. The data of sales have been collected on these 16 combinations from two retail stores during a month period as given below.

**Table-1:** Sample Data

The hypotheses for the analysis are:

**Null hypothesis-**

H_{01}: The average sale remains the same for three types of flavour. (First main effect)

H_{02}: The average sale remains the same for all levels of brand. (Second main effect)

H_{03}: The average sale remains the same for all combinations of flavour and brand. (Interaction effect)

**Alternative Hypothesis- **

H_{11}: The average sale differs for at least two types of flavour.

H_{12}: The average sale differs for at least two all levels of brand.

H_{13}: The average sale differs for at least two combinations of flavour and brand combination.

**Table-2:** Input Data

**Performing the Analysis with SPSS**

For SPSS Version 11, click on **Analyze ⇒ ****General Linear Model ⇒ Univariate. **This will bring up the SPSS screen dialogue box as shown below.

After clicking **Univariate, **this will bring up the following SPSS screen dialogue box.

Select the dependent variable “flavour” and click it to move to **Dependent variables** box. Select the independent variables ‘brand’ and ‘sale’ and move them to **Fixed Factor(s) Box.**

Then click **Model** followed by **Full Factorial** and this will bring up the following dialogue box.

Then click **Continue** to return the main dialogue box and click OK to get the output.

**SPSS Output**

The SPSS output of the analysis is given in the following tables.

**Univariate Analysis of Variance**

**Table-3:** Between-Subjects Factors

Tests of Between-Subjects Effects

**Table-4:** Dependent Variable: Sales

*a R Squared = .465 (Adjusted R Squared = -.037)*

** **

**From the output, F = 0.466 (Flavour), **F** = 0.397 (Brand), F = 1.256 (Flavour and Brand)**

**DECISION**

Reject the null hypothesis if p-value (Sig. value) ≤ 0.05

**INTERPRETATION**

- The p-value for flavor is 0.710 and it is more than 05 (
**5% level of significance**), so we accept the null hypothesis and conclude that flavor has no effect on sales. - The p-value for flavor is 0.757 and it is more than 05 (
**5% level of significance**) so we accept the null hypothesis at 5% level of significance. It can be concluded that brand has no effect on sales. - The p-value for flavor and brand is 0.331 and it is less than05
**(5% level of significance**) so we reject the null hypothesis at 5% level of significance and conclude that the interaction effect of flavor and brand is insignificant.

**CASE ANALYSIS-2 **

** ****PROBLEM**

The following data set refers to the salary package (in lakhs) offered to MBA graduates with different specialization selected from three different pattern of studies. Three students have been selected from each pattern of studies with the same specialization as given below.

**Table-1:** Sample Data

The interest is to test whether there is any effect of specialization (main effect due to first independent variable) ,the pattern of studies (main effect due to second independent variable) and the combined effect of pattern of studies and the specialization (interaction effect) on the salary package (dependent variable) or not.

**Code:**

The hypotheses for the analysis are:

**Null hypothesis-**

H_{01}: The average salary package offered remains the same for three types of specialization. (First main effect)

H_{02}: The average salary package offered remains the same for three patterns of studies. (Second main effect)

H_{03}: The average salary package offered remains the same for specialization and pattern of studies. (Interaction effect)

**Alternative Hypothesis- **

H_{11}: The average salary package differs for at least two types of specialization.

H_{12}: The average salary package differs for at least two levels of patterns of studies.

H_{13}: The average salary package differs for at least two combinations of specialization and pattern of studies.

**Table-2:** Input Data

**SPSS Output**

The SPSS outputs are illustrated in below tables.

**Univariate Analysis of Variance**

**Table-2:** Between-Subjects Factors

*Tests of Between-Subjects Effects*

**Table-3:**Dependent Variable: Salary Package

*a R Squared = .759 (Adjusted R Squared = .652)*

** ****From the output, F = 2.404 (Specialization), F = 24.593 (Pattern of studies), F = 0.667 (Specialization * Pattern of studies)**

**DECISION**

Reject the null hypothesis if p-value (Sig. value) ≤ 0.05

**INTERPRETATION**

- The p-value for Specialization is 0.119 and it is more than 05 (
**5% level of significance**), so we accept the null hypothesis and conclude that flavor has no effect on salary package. - The p-value for flavor is 0.000 and it is less than 05 (
**5% level of significance**) so we reject the null hypothesis at 5% level of significance. It can be concluded that salary package is affected by the pattern of studies. - The p-value for flavor and brand is 0.623 and it is more than05
**(5% level of significance**) so we accept the null hypothesis at 5% level of significance and conclude that the interaction effect of specialization and pattern of studies has no effect on salary package.

**SPSS Command**

- Click on ANALYZE at the SPSS menu bar (in older versions of SPSS, click on STATISTICS instead of ANALYZE).
- Click on GENERAL LINEAR MODEL followed by UNIVARIATE.
- Select the appropriate variable and move it to the DEPENDENT LIST. Select the fixed factors and move them to FIXED FACTOR(S) BOX.
- Then click MODEL followed by FULL FACTORIAL.
- Click CONTINUE to return the main dialogue box.
- Finally click OK to get the output for FACTORIAL DESIGN.