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Srinivasulu Rajendran
 Centre for the Study of Regional Development (CSRD)


Jawaharlal Nehru University (JNU)
                      New Delhi
                        India
              r.srinivasulu@gmail.com
Objective of the session



      To understand two-way
      anova through software
             packages
1. What is the procedure to
perform 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 to
run a two-way ANOVA
Testing of Assumptions
 In SPSS, homogeneity of variances is tested using
 Levene's 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.
STEP 1
Click Analyze > General Linear Model > Univariate...
on the top menu as shown below
STEP 2
You will be presented with the "Univariate" dialogue box:
STEP 3
 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)”
STEP 4
Click on the Plot button. You will be presented with the
"Univariate: Profile Plots" dialogue box
STEP 5
 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.
STEP 8
Click the “Post Hoc..” button. You will be presented with the
  "Univariate: Post Hoc Multiple Comparisons for Observed..."
  dialogue box as shown below:
STEP 9
 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
Topic 14 two anova
STEP 10
 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.
Topic 14 two anova
STEP 11
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 provides
the mean and standard deviation for
the groups that have been split by
both independent variables. In
addition, the table also provides
"Total" rows, which allows means
and standard deviations for groups
only split by one independent
variable or none at all to be known.
From this table we can
                                  Levene's Test of Equality of Error Variancesa
see that we don’t have
homogeneity            of
variances      of     the
dependent        variable     Dependent Variable:Per capita monthly food expenditure
across groups. We                                     (taka)
know this as the Sig.
value is less than 0.05,           F            df1           df2           Sig.
                                 2.335           6           1228           .030
which is the level we
set for alpha. So we
have concluded that
the variance across
                            Tests the null hypothesis that the error variance of the
groups               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             1235

Corrected Total    333066026              1234

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Topic 14 two anova

  • 1. Srinivasulu Rajendran Centre for the Study of Regional Development (CSRD) Jawaharlal Nehru University (JNU) New Delhi India r.srinivasulu@gmail.com
  • 2. Objective of the session To understand two-way anova through software packages
  • 3. 1. What is the procedure to perform Two-way ANOVA? 2. How do we interpret results?
  • 4. 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 .
  • 5. 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.
  • 6.  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".
  • 7. How to correctly enter your data into SPSS in order to run a two-way ANOVA
  • 8. Testing of Assumptions  In SPSS, homogeneity of variances is tested using Levene's 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.
  • 10. Click Analyze > General Linear Model > Univariate... on the top menu as shown below
  • 12. You will be presented with the "Univariate" dialogue box:
  • 14.  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)”
  • 16. Click on the Plot button. You will be presented with the "Univariate: Profile Plots" dialogue box
  • 18.  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.]
  • 19. STEP 6 & 7
  • 20.  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.
  • 22. Click the “Post Hoc..” button. You will be presented with the "Univariate: Post Hoc Multiple Comparisons for Observed..." dialogue box as shown below:
  • 24.  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
  • 27.  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.
  • 30. Click the “Ok” button to generate the output.
  • 31. SPSS Output of Two-way ANOVA
  • 32.  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
  • 33. This table is very useful as it provides the mean and standard deviation for the groups that have been split by both independent variables. In addition, the table also provides "Total" rows, which allows means and standard deviations for groups only split by one independent variable or none at all to be known.
  • 34. From this table we can Levene's Test of Equality of Error Variancesa see that we don’t have homogeneity of variances of the dependent variable Dependent Variable:Per capita monthly food expenditure across groups. We (taka) know this as the Sig. value is less than 0.05, F df1 df2 Sig. 2.335 6 1228 .030 which is the level we set for alpha. So we have concluded that the variance across Tests the null hypothesis that the error variance of the groups was dependent variable is equal across groups. significantly different a. Design: Intercept + head_sex + head_edu + (unequal). head_sex * head_edu
  • 35. 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).
  • 36. 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 1235 Corrected Total 333066026 1234