Srinivasulu Rajendran Centre for the Study of Regional Development (CSRD)Jawaharlal Nehru University (JNU) New Delhi India firstname.lastname@example.org
Objective of the session To understand two-way anova through software packages
1. What is the procedure toperform Two-way ANOVA?2. How do we interpret results?
Two-way ANOVA using SPSS The two-way ANOVA compares the mean differences between groups that have been split on two independent variables (called factors). You need two independent, categorical variables and one continuous, dependent variable .
Objective We are interested in whether an monthly per capita food expenditure was influenced by their level of education and their gender head. Monthly per capita food expenditure with higher value meaning a better off. The researcher then divided the participants by gender head of HHs i.e Male head & Female head HHs and then again by level of education.
In SPSS we separated the HHs into their appropriate groups by using two columns representing the two independent variables and labelled them “Head_Sex" and “Head_Edu". For “head_sex", we coded males as "1" and females as “0", and for “Head_Edu", we coded illiterate as "1", can sign only as "2" and can read only as "3“ and can read & write as “4”. Monthly per capita food expenditure was entered under the variable name, “pcmfx".
How to correctly enter your data into SPSS in order torun a two-way ANOVA
Testing of Assumptions In SPSS, homogeneity of variances is tested using Levenes Test for Equality of Variances. This is included in the main procedure for running the two- way ANOVA, so we get to evaluate whether there is homogeneity of variances at the same time as we get the results from the two-way ANOVA.
Click Analyze > General Linear Model > Univariate...on the top menu as shown below
You will be presented with the "Univariate" dialogue box:
You need to transfer the dependent variable “pcmfx" into the "Dependent Variable:" box and transfer both independent variables, “head_sex" and “head_edu", into the "Fixed Factor(s)”
Click on the Plot button. You will be presented with the"Univariate: Profile Plots" dialogue box
Transfer the independent variable “head_edu" from the "Factors:" box into the "Horizontal Axis:" box and transfer the “head_sex" variable into the "Separate Lines:" box. You will be presented with the following screen: [Tip: Put the independent variable with the greater number of levels in the "Horizontal Axis:" box.]
STEP 6 & 7
Click the “add” button You will see that “head_edu*head _sex" has been added to the "Plots:" box. Click the “continue” button. This will return you to the "Univariate" dialogue box.
Click the “Post Hoc..” button. You will be presented with the "Univariate: Post Hoc Multiple Comparisons for Observed..." dialogue box as shown below:
Transfer “head_edu" from the "Factor(s):" box to the "Post Hoc Tests for:" box. This will make the "Equal Variances Assumed" section become active (loose the "grey sheen") and present you with some choices for which post-hoc test to use. For this example, we are going to select "Tukey", which is a good, all-round post-hoc test. [You only need to transfer independent variables that have more than two levels into the "Post Hoc Tests for:" box. This is why we do not transfer “head_sex".] You will finish up with the following screen Click the “Continue” button to return to the "Univariate" dialogue box
Click the “option” button. This will present you with the "Univariate: Options" dialogue box as shown below: Transfer “head_sex", “head_edu" and “head_sex*head_edu" from the "Factor(s) and "Factor Interactions:" box into the "Display Means for:" box. In the "Display" section, tick the "Descriptive Statistics" and "Homogeneity tests" options. You will presented with the following screen Click the “continue” button to return to the "Univariate" dialogue box.
Click the “Ok” button to generate the output.
SPSS Output of Two-way ANOVA
SPSS produces many tables in its output from a two-way ANOVA and we are going to start with the "Descriptives" table as shown below: Descriptive Statistics Dependent Variable:Per capita monthly food expenditure (taka) Head of the Household - Sex (sum) head_edu Mean Std. Deviation N Male 1 939.8895 455.16118 245 2 998.0697 491.73339 262 3 858.3107 383.20545 20 4 1137.9562 534.76858 571 Total 1055.2881 512.60856 1098 Female 1 962.6195 627.75916 44 2 967.0070 424.26461 41 4 1205.5084 607.04529 52 Total 1056.1239 574.00781 137 Total 1 943.3501 484.17553 289 2 993.8665 482.62690 303 3 858.3107 383.20545 20 4 1143.5946 540.95653 623 Total 1055.3809 519.52636 1235
This table is very useful as it providesthe mean and standard deviation forthe groups that have been split byboth independent variables. Inaddition, the table also provides"Total" rows, which allows meansand standard deviations for groupsonly split by one independentvariable or none at all to be known.
From this table we can Levenes Test of Equality of Error Variancesasee that we don’t havehomogeneity ofvariances of thedependent variable Dependent Variable:Per capita monthly food expenditureacross groups. We (taka)know this as the Sig.value is less than 0.05, F df1 df2 Sig. 2.335 6 1228 .030which is the level weset for alpha. So wehave concluded thatthe variance across Tests the null hypothesis that the error variance of thegroups was dependent variable is equal across groups.significantly different a. Design: Intercept + head_sex + head_edu +(unequal). head_sex * head_edu
Tests of Between-Subjects Effects Table The table shows the actual results of the two-way ANOVA as shown We are interested in the head of hhs gender, education and head_sex*head_edu rows of the table as highlighted above. These rows inform us of whether we have significant mean differences between our groups for our two independent variables, head_sex and head_edu, and for their interaction, head_sex*head_edu. We must first look at the head_sex*head_edu interaction as this is the most important result we are after. We can see from the Sig. column that we have a statistically NOT significant interaction at the P = .686 level. You may wish to report the results ofhead_sex and head_edu as well. We can see from the above table that there was no significant difference in monthly per capita food exp between head_sex (P = .675) but there were significant differences between educational levels (P < .000).
Tests of Between-Subjects Effects Dependent Variable:Per capita monthly food expenditure (taka) Type III Sum of Source Squares df Mean Square F Sig.Corrected Model 10669432 6 1778239 6.773 .000 Intercept 279013110 1 279013110 1062.753 .000 head_sex 46145 1 46145 .176 .675 head_edu 5527869 3 1842623 7.019 .000 head_sex * 197900 2 98950 .377 .686 head_edu Error 322396593 1228 262538 Total 1708644528 1235Corrected Total 333066026 1234