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Parametric Comparative Statistics
Introduction
In this chapter you will perform parametric comparative statistical tests
using SPSS. A step by step SPSS procedure with pictures for each test
was provided.
Intended Learning Outcomes
At the end of this chapter, you will be able to:
- demonstrate understanding of the comparative statistical tests;
- perform comparative statistical analysis using SPSS; and
- present and interpret the results of the analysis in APA style. 1. Independent samples t-test
Independent samples t-test tells you whether there is a statistically
significant difference in the mean scores for two groups. In statistical
terms, you are testing the probability that the two sets of scores came from
the same population. In this test, you need one independent variable with
two categories (e. g. Class types: Science class or Regular class) and one
continuous dependent variable (e. g. Math test scores). The nonparametric
alternative for Independent samples t-test is the Mann Whitney U-test.
Example 7.
Suppose you want to explore whether there is a significant difference
between the academic performance of Science class and Regular class
students. You administered the same test to 14 randomly selected
students of each group. The test scores of the students are shown below:
Science Class 32 38 37 36 36 34 39 36 37 42 38 38 36 35
Regular Class 30 36 32 34 31 30 34 33 34 35 35 36 32 30
How to create SPSS data file for Independent samples t-test?
- Open SPSS. Click Variable View. Encode the independent variable
(e. g. Type of Class) and the dependent variable (e. g. Test scores) in
the column Label. Code the categories of your independent variable
in the Values column. In this example, we let 1.00 = Science Class
and 2.00 = Regular Class.
- Go to Data View. Remember that there are 14 Science class
students (coded 1) and 14 Regular class students (coded 2). So we
input fourteen 1’s and fourteen 2’s in column ‘ClassType’ with their
corresponding test scores in column ‘TestScores’.
Procedures in Requesting for Independent samples t-test
- Click on Analyze , Compare Means , and then Independent
Samples T Test.
Interpretation of the Outputs from Independent samples t-test
- The Group Statistics table shows the results of the descriptive
statistics. The two means we are comparing, 36.71 and 33.00, differs
by 3.71. But do we have sufficient evidence to say that the difference
of 3.71 is significant when the standard deviations are considered?
This question is answered in the next table.
Group Statistics
Type of Class N Mean Std. Deviation Std. Error Mean
Test Scores Science Class 14 36.7143 2.36736.
Regular Class 14 33.0000 2.18386.
- The Independent Samples Test table contains the results of the
Levene’s test and the t-test. The Levene’s test checks whether the
assumption for homogeneity of variances is met. Since the sig. value
of .777 is greater than .05, then the assumption is not violated.
Therefore, we will use the t-test results for the row of ‘Equal
variances assumed’. As shown, the sig. (probability) value of the t
value (4.315) is less than. 01 which suggests that the mean
difference of 3.71 is highly significant. Statistically speaking, we
have 99% confidence level (sufficient evidence) in claiming that the
two groups differ in terms of their test performance.
Independent Samples Test
Levene's Test for Equality of Variances t-test for Equality of Means
F Sig. t df
Sig. (2- tailed)
Mean Difference
Std. Error Difference
95% Confidence Interval of the Difference
Lower Upper
Test Scores
Equal variances assumed
.082 .777 4.315 26 .000 3.7142 .860 1.944 5.
Equal variances not assumed
4.315 25.83 .000 3.7142 .860 1.944 5.
- The effect size is not available in both tables. But we can use the
Partial Eta Squared formula and calculate for the effect size as
follows:
2
2
Based on the Cohen’s guidelines, an eta squared of .01 is small ,
.06 is moderate , and .14 is large. Hence, the above-computed 0.
is a very large effect size.
Presenting the Results in APA style
Independent samples t-test was conducted to compare the test scores for
Science class and Regular class. Results shown in Table 1 indicated a
significant difference between the performance of Science class students
( M = 36.71; SD = 2.36) and Regular class students ( M = 33; SD = 218), t
(26) = 4.32, p < .01, two-tailed. The magnitude of the mean difference (3.71;
95% CI: 1.94 to 5.48) was very large,
2
Table 1
Test performance of Science and Regular class students
Type of Class n M SD t(26) p ^2 95% CI
Science 14 36.71 2.36 4.315 .00 0.42 [1.94,
5.48]
Regular 14 33.00 2.
Activity 17
In a Statistics training program, 12 trainees were taught in data analysis
using Statistical Package 1 (SP1) and another group of 13 trainees were
exposed to data analysis through the use of a different Statistical Package,
SP2. The same data analysis questions were administered to measure their
achievement and their scores are shown in the following table. Is there a
significant difference between the achievements of the two groups of
trainees? Report the results in APA style.
SP 1 81 71 79 83 76 75 84 90 83 78 84 83
SP 2 69 75 72 69 67 74 70 66 76 72 88 84 86
Procedures for requesting Paired samples t-test
- Click on Analyze , Compare Means , and then Paired Samples T
Test.
- Click on the two variables (e. g. Pretest and Posttest) and move
them into the box labeled Paired Variables. Then click OK.
Interpreting the Outputs generated
- The Paired Samples Statistics table shows the results of
descriptive statistics. The difference between their Mean Pretest
score ( M = 51.30) and Mean Posttest score ( M = 57.60) is −6.30.
Paired samples t-test will detect if this difference is significant or
not.
Paired Samples Statistics
Mean N Std. Deviation Std. Error Mean
Pair 1 Pretest 51.30 10 15.853 5.
Posttest 57.60 10 17.589 5.
- The Paired Samples Test table shows a probability (sig.) value of
.016 which means that the mean difference is significant at .05 level.
Statistically speaking, there is 95% confidence level that the
trainees’ pretest and posttest scores are significantly different.
Paired Samples Test
Paired Differences
Mean SD Std. Error Mean T df Sig. (2-tailed)
95% Confidence Interval of the Difference
Lower Upper
Pair 1 Pretest-Posttest - 6.30 6.733 2.129 - 11.117 - 1.482 - 2.959 9.
- The effect size can be calculated using Partial Eta Squared as
follows:
2
2 2
Based on the guidelines proposed by Cohen (1988), the effect size
of 0.49 is very large.
Presenting the results in APA style
Paired samples t-test was run to compare the writing proficiency scores of
the trainees before and after they were exposed to academic writing
training. Results showed a significant increase in their writing proficiency
from pretest ( M = 51.30; SD = 15.85) to posttest ( M = 57.60; SD = 17.58), t
(9) = −2.95, p < .05, two-tailed. The mean increase of 6.30 was large,
2 =
0.49, 95% CI from −11.11 to −1.48.
Table 2
Pretest and posttest performance of the trainees
Test M SD t (9) p
2
95% CI
Pretest 51.30 15.85 - 2.95 .016 0.49 [-11.11, - 1.48]
Posttest 57.60 17.
Activity 18
The following data represent the pre-test and post-test scores of 10
randomly selected individuals in entrepreneurship training. Do the
differences in the scores of the participants suggest that the training was
effective? Present the results in APA style.
Pre-test 40 30 48 45 70 65 30 50 60 75
- Go to Data View. For organization purpose, encode first the data of
the 6 students exposed to Method A, followed by those 6 students to
Method B, and then to Method C.
Procedure in Requesting for One-way ANOVA
- Click on Analyze , Compare Means , and then One-way ANOVA.
- Click on your dependent (continuous) variable (e. g. English
Proficiency) and move it into the box marked Dependent list. Click
on your independent (categorical) variable (e. g. Teaching Method
with subgroups Method A, Method B, and Method C). Move it into
the box labeled Factor. Click on Options button.
- Tick Descriptive and Homogeneity of variance test. Click on
Continue and then OK.
Interpreting the Results of the ANOVA
- The Descriptives table apparently shows that there are three
different mean scores. However, it does not show whether the
difference between these means is significant or not.
LSD is not conservative and too statistically powerful whose results
tend to commit Type I error (rejection of the null hypothesis when it
should be accepted).
Bonferonni is a simple procedure but in most cases not powerful
(Day & Quinn, 1989). Results from both Scheffe and Bonferonni
likely lead to Type II error (acceptance of the null hypothesis when
it should be rejected).
Tukey’s procedure is the best for all possible pairwise comparisons
for any kind of sample sizes (equal/unequal sample sizes
with/without confidence intervals). Both Stevens (1999) and Day
and Quinn (1989) agree that Tukey is the procedure of choice when
all means are being compared.
Procedure for requesting Tukey
- Click again on Analyze , Compare Means , and One way ANOVA.
Then click Post Hoc button.
- Tick Tukey and then Continue. Finally, click OK.
Interpreting the result of Post hoc analysis using Tukey
- The Multiple Comparisons table shows the detailed comparison of all
pairs of groups. It gives you an idea about the probability (sig.) value
and confidence interval for each comparison. The sig. or probability
value of .324 shows that Method A and Method B are not
significantly different; p = .001 shows that Method A and Method C
are significantly different; while p = .010 denotes that Method B and
Method C are significantly different. However, the next table gives
you a quicker way of determining which groups are significantly
different (and not significantly different).
Multiple Comparisons
English Proficiency Score Tukey HSD
(I) Method (J) Method
Mean Difference (I-J) Std. Error Sig.
95% Confidence Interval
Lower Bound Upper Bound
Method A Method B 4.83333 3.24665 .324 - 3.5997 13.
Method C 16.00000*^ 3.24665 .001 7.5669 24.
Method B Method A - 4.83333 3.24665 .324 - 13.2664 3.
Method C 11.16667*^ 3.24665 .010 2.7336 19.
Method C Method A - 16.00000*^ 3.24665 .001 - 24.4331 - 7.
Method B - 11.16667*^ 3.24665 .010 - 19.5997 - 2.
*. The mean difference is significant at the 0.05 level.
- The following table on English Proficiency Score summarizes the
results of the Multiple Comparisons table. It directly shows which
methods are significantly different and which are not. Methods that
belong to the same column are not significantly different while
Methods that belong to different columns are significantly different.
As shown in the table, Method A and Method B are the not
significantly different groups but both are significantly different to
Method C.
English Proficiency Score
Tukey HSD
Method N
Subset for alpha = 0.
1 2
Method C 6 77.
Method B 6 88.
Method A 6 93.
Sig. 1.000.
Means for groups in homogeneous subsets are displayed.
4. Two-way Analysis of Variance
Two-way ANOVA allows you to simultaneously test for the effect of each of
your independent variables on the dependent variable and also identifies
any interaction effect between the independent variables. To do Two-way
ANOVA, you need two categorical independent variables and one
continuous dependent variable.
Example 7.
Suppose you wanted to investigate the impact of exposure to reach-out
activity (exposed or not exposed) and socio-economic status (high, average,
or low) on social responsibility scores of 24 randomly selected teachers.
The data you collected are given in the table below:
Exposure to
reach-out activity
Socio-Economic Status
High Average Low
Exposed 15 8 8
Not Exposed 15 8 2
6 10 4
How to create the SPSS data file?
- Open SPSS. Go to Variable View. Include ID (identification) to check
for the data of each subject. Encode the three variables in the Name
and Label column.
- Code the two categorical variables ( SES and Exposure ) in the Values
section.
- Go to Data View and input the data. There must be 24 rows as there
are 24 subjects. Encode all the data per subject. Utilize the values
used in coding the categories of the categorical variables.
Procedures for Two-way ANOVA
- Click on Analyze , General Linear Model , and then on Univariate.
Interpreting the results of Two-way ANOVA
- The Between-Subjects Factors table shows the number of cases
of every subgroup of the variables. It shows that there are 8 cases
for each of the three SES subgroups and 12 cases for each of the
two subgroups of Exposure to Reach-out Activity.
- The Descriptive Statistics table gives information about the Mean
(social responsibility) scores being compared. It appears in the table
that those who were both exposed to reach-out activities and belong
to high SES had higher Mean social responsibility scores. You must
see the result of the ANOVA to validate whether the differences of
the mean scores are really significant.
- The Levene’s Test of Equality of Error Variances checks whether
the assumption on the homogeneity of variances is violated or met.
Since the test statistic F = 1.33 was not significant, p > .29, the data
met the said parametric assumption.
- The Tests of Between-Subjects Effects table shows the result of
the ANOVA. Proceed directly to the probability (sig.) values of the F
values of SES , Exposure and SESExposure*. For SES , the F
statistic of 9.16 is significant at .01 level which means that there is
99% confidence level in claiming that at least one of the three groups
of SES had significantly different Mean social responsibility score.
For Exposure , notice that F = 8.40 is significant at .05 (but not at
.01 level as there are still smaller decimals not shown in the table)
which suggests a significant difference in the mean social
responsibility scores between those exposed to reach-out activities
and not exposed to reach-out activities. Moreover, for
SESExposure* , the F = .229 is not significant (p > .79) which
indicates that there is no interaction between the SES and Exposure
to reach-out activities in terms social responsibility scores.
- The Multiple Comparisons table shows which pair(s) of SES groups
had significantly different social responsibility scores. The results
showed that there is a significant difference between the social
responsibility scores between Low SES and Average SES ( p < .05)
and also between Low SES and High SES ( p < .01). Note that there
is no Multiple Comparisons table for Exposure variable because it
has only two subgroups (exposed and not exposed).
- The results of the Multiple Comparisons table, however, can be
more easily visualized in the Homogenous table below. As shown,
only the Low SES belongs to the Subset 1 which means that it has
significantly lower mean social responsibility score as compared to
mean social responsibility scores of Average SES and High SES
which are both in Subset 2.
