Transcript SW388R6 Data Analysis and Computers I Slide 1 Independent Samples T-Test of Population Means Key Points about Statistical Test Sample Homework Problem Solving the Problem with SPSS Logic for.

SW388R6
Data Analysis
and Computers I
Slide 1
Independent Samples T-Test
of Population Means
Key Points about Statistical Test
Sample Homework Problem
Solving the Problem with SPSS
Logic for Independent Samples T-Test
of Population Means
Power Analysis
SW388R6
Data Analysis
and Computers I
Independent Samples T-Test: Purpose
Slide 2


Purpose: test whether or not the populations
represented by the two samples have a different
mean
Examples:



Social work students have higher GPA’s than nursing students
Social work students volunteer for more hours per week
than education majors
UT social work students score higher on licensing exams
than graduates of Texas State University
SW388R6
Data Analysis
and Computers I
Independent Samples T-Test: Hypotheses
Slide 3

Hypotheses:





Null: mean of population 1 = mean of population 2
Versus
Research: mean of population 1 < mean of population 2
Research: mean of population 1 ≠ mean of population 2
Research: mean of population 1 > mean of population 2
Decision:


Reject null hypothesis if pSPSS ≤ alpha (≠ relationship)
Reject null hypothesis if pSPSS÷2 ≤ alpha (< or > relationship)
SW388R6
Data Analysis
and Computers I
Slide 4
Independent Samples T-Test: Assumptions and
Requirements



Variable is interval level (ordinal with caution)
Variable is normally distributed
 Acceptable degree of skewness and kurtosis
or
 Using the Central Limit Theorem (30+ in each
group)
The variance of the two groups is not different (if
different, use alternative formula)
SW388R6
Data Analysis
and Computers I
Independent Samples T-Test: Effect Size
Slide 5



Cohen’s d measures difference in means in standard
deviation units.
Cohen’s d = difference in population means
population standard deviation
Interpretation:
 small: d = .20 to .50
 medium: d = .50 to .80
 large: d = .80 and higher
SW388R6
Data Analysis
and Computers I
Independent Samples T-Test: APA Style
Slide 6

An independent samples T-test is presented the same
as the one-sample t-test:
t(75) = 2.11, p = .02 (one –tailed), d = .48
Degrees
of
freedom

Value
of
statistic
Significance
of statistic
Include if test
is one-tailed
Effect size
if available
Example: Survey respondents who were employed by
the federal, state, or local government had
significantly higher socioeconomic indices (M =
55.42, SD = 19.25) than survey respondents who were
employed by a private employer (M = 47.54, SD =
18.94) , t(255) = 2.363, p = .01 (one-tailed).
Homework problems: Independent Samples
T-Test of Population Means
SW388R6
Data Analysis
and Computers I
Slide 7
This problem uses the data set GSS2000R.Sav to compare the average score on the variable
"highest year of school completed" [educ] for groups of survey respondents defined by the
variable "governmental employment" [wrkgovt]. Using an independent samples t-test with
an alpha of .05, is the following statement true, true with caution, false, or an incorrect
application of a statistic?
Survey respondents who were employed by the federal, state, or local government
completed significantly more years of school (M = 13.97, SD = 3.27) than survey
respondents who were employed by a private employer (M = 13.07, SD = 2.84) .
o
o
o
o
True
True with caution
False
Incorrect application of a statistic
This is the general framework
for the problems in the
homework assignment on
“Independent Samples T-Test of
Population Means.” The
description is similar to findings
one might state in a research
article.
Homework problems: Independent Samples
T-Test - Data set, variables, and sample
SW388R6
Data Analysis
and Computers I
Slide 8
This problem uses the data set GSS2000R.Sav to compare the average score on the variable
"highest year of school completed" [educ] for groups of survey respondents defined by the
variable "governmental employment" [wrkgovt]. Using an independent samples t-test with
an alpha of .05, is the following statement true, true with caution, false, or an incorrect
application of a statistic?
Survey respondents who were employed by the federal, state, or local government
first
paragraph
completed significantly The
more
years
of schoolidentifies:
(M = 13.97, SD = 3.27) than survey
respondents who were employed
by a set
private
employer
(M = 13.07, SD = 2.84) .
• The data
to use,
e.g. GSS2000R.Sav
o
o
o
o
• The groups that will be compared in the
analysis
• The variable compared in the t-test
• The alpha level to use for the hypothesis test
True
True with caution
False
Incorrect application of a statistic
Homework problems: Independent Samples
T-Test - Specifications
SW388R6
Data Analysis
and Computers I
Slide 9
This problem uses the data set GSS2000R.Sav to compare the average score on the variable
"highest year of school completed" [educ] for groups of survey respondents defined by the
variable "governmental employment" [wrkgovt]. Using an independent samples t-test with
an alpha of .05, is the following statement true, true with caution, false, or an incorrect
application of a statistic?
Survey respondents who were employed by the federal, state, or local government
completed significantly more years of school (M = 13.97, SD = 3.27) than survey
respondents who were employed by a private employer (M = 13.07, SD = 2.84) .
o
o
o
o
True
True with caution
False
Incorrect application of a statistic
The second paragraph specifies:
• The sample means and standard
deviation for the groups being
compared
• The relationship for deriving the
research hypothesis
Homework problems: Independent Samples
T- Test - Choosing an answer
SW388R6
Data Analysis
and Computers I
Slide 10
The answer to a problem
will be True if the t-test
The
answerthe
to a
problem
willon
bethe variable
This
problem
the data
set GSS2000R.Sav to
compare
average
score
supports
theuses
finding
in
True with caution if the t-test
the problem
statement.
"highest
year of
school completed" [educ] for groups
of survey
respondents
supports
the finding
in the defined by the
problem
statement,
but samples
the
variable "governmental employment" [wrkgovt].
Using an
independent
t-test with
dependent variable is ordinal level.
an alpha of .05, is the following statement true, true with caution, false, or an incorrect
application of a statistic?
Survey respondents who were employed by the federal, state, or local government
completed significantly more years of school (M = 13.97, SD = 3.27) than survey
respondents who were employed by a private employer (M = 13.07, SD = 2.84) .
o
o
o
o
True
True with caution
False
Incorrect application of a statistic
The answer to a
problem will be
False if the t-test
does not support the
finding in the
problem statement.
The answer to a problem will Incorrect
application of a statistic if
• the t-test violates the level of
measurement requirement, i.e. the
dependent variable is nominal
• the assumption of normality of the
dependent variable is violated and
the central limit theorem doesn’t
apply
• the independent variable is not
dichotomous
SW388R6
Data Analysis
and Computers I
Slide 11
Solving the problem with SPSS:
Identifying numeric codes for groups - 1
Our first task in SPSS is to
identify the numeric codes
for the groups that SPSS
will require us to specify.
The problem statement
tells us “This problem uses
the data set GSS2000R.Sav
to compare the average
score on the variable
"highest year of school
completed" [educ] for
groups of survey
respondents defined by the
variable "governmental
employment" [wrkgovt].”
Select the
Variables… command
from the Utilities
menu.
NOTE: in our problems we required that the
grouping, or independent variable, be dichotomous,
because there are other statistical tests to use
when there are more than two groups. SPSS does
not require the independent variable to be
dichotomous, but it does require that you enter the
numeric codes for the two groups (possibly out of a
larger number of groups) that you wish to
compare.
SW388R6
Data Analysis
and Computers I
Slide 12
Solving the problem with SPSS:
Identifying numeric codes for groups - 2
Scroll through the list
of variables until you
see wkgovt. Click on
wkgovt and the
information for the
variable appears in
the panel to the
right.
Click on
Close to
dismiss the
dialog box.
The Variable Information panel shows us
the text labels that the creator of the data
set assigned to each of the possible numeric
responses for this variable.
The numeric codes for the groups we
want to compare are: 1 (GOVERNMENT)
and 2 (PRIVATE).
This
remaining
numeric codes
represent
missing data:
0 (NAP),
8 (DK), and
9 (NA).
SW388R6
Data Analysis
and Computers I
Slide 13
Solving the problem with SPSS:
Level of measurement
Statistical tests of means require that the
dependent variable be interval level. "Highest
year of school completed" [educ] is interval
level, satisfying the requirement.
In our analyses, we will allow the dependent
variable to be ordinal , which violates this
requirement in the strictest interpretation of
level of measurement. However, since the
research literature often computes means for
ordinal level data, especially scaled measures,
we will follow the convention of applying
interval level statistics to ordinal data. Since all
analysts may not agree with this convention, a
caution is added to any true findings.
SW388R6
Data Analysis
and Computers I
Slide 14
Solving the problem with SPSS:
Evaluating normality - 1
The independent samples ttest uses the t-distribution for
the probability of the test
statistic. To obtain accurate
probabilities, the variable must
follow a normal distribution.
We will generate descriptive
statistics to evaluate
normality.
Select the Descriptive
Statistics > Descriptives…
command from the Analysis
menu.
SW388R6
Data Analysis
and Computers I
Slide 15
Solving the problem with SPSS:
Evaluating normality - 2
First, move the
variable we will use in
the t-test, educ, to the
Variable(s) list box.
Second, click on
the Options…
button to select
the statistics we
want.
SW388R6
Data Analysis
and Computers I
Slide 16
First, in addition
to the statistics,
SPSS has checked
by default, mark
the Kurtosis and
Skewness check
boxes on the
Distribution panel.
Solving the problem with SPSS:
Evaluating normality - 3
Second, click on the
Continue button to
close the dialog box.
SW388R6
Data Analysis
and Computers I
Slide 17
Solving the problem with SPSS:
Evaluating normality - 4
Click on the OK
button to obtain
the output.
SW388R6
Data Analysis
and Computers I
Slide 18
Solving the problem with SPSS:
Evaluating normality - 5
"Highest year of school completed"
[educ] did not satisfy the criteria for a
normal distribution. The skewness of the
distribution (-.137) was between -1.0
and +1.0, but the kurtosis of the
distribution (1.246) fell outside the
range from -1.0 to +1.0.
Having failed the normality requirement
using this criteria, we will see if we can
apply the central limit theorem.
SW388R6
Data Analysis
and Computers I
Slide 19
Solving the problem with SPSS:
The independent-samples t-test - 1
The number of cases in
each group is part of the
output for the independent
samples t-test, so we will
go ahead and compute that
test to continue addressing
the issue of normality.
Select Compare Means
> IndependentSamples T Test… from
the Analyze menu.
SW388R6
Data Analysis
and Computers I
Slide 20
Solving the problem with SPSS:
The independent-samples t-test - 2
First, move the
dependent variable educ
to the Test Variable(s)
list box.
Second, move the
independent variable
wkgovt to the Grouping
Variable text box.
Note that SPSS lists two
question marks after the
variable name and activates
the Define Groups… button
as its clue for what it wants
us to do next. Click on the
Define Groups button.
SW388R6
Data Analysis
and Computers I
Slide 21
Solving the problem with SPSS:
The independent-samples t-test - 3
First, type in the
numeric codes for the
groups in the wkgovt
variable that we looked
up at the beginning of
the problem.
Second, click on the
Continue button to
close the dialog box.
SW388R6
Data Analysis
and Computers I
Slide 22
Solving the problem with SPSS:
The independent-samples t-test - 4
Click on the OK
button to close
the dialog box.
Note that SPSS has
replaced the question
marks after the variable
name with the numeric
codes we typed in.
SW388R6
Data Analysis
and Computers I
Slide 23
Solving the problem with SPSS: Evaluating
normality with the central limit theorem - 6
Since survey respondents who were employed by
the federal, state, or local government had 38 cases
and survey respondents who were employed by a
private employer had 217 cases, the assumption of
normality was satisfied by the Central Limit
Theorem which required both groups to have 30 or
more cases.
If we are unable to establish
normality either by the
distribution or by the central
limit theorem, the t-test would
not be an appropriate statistic.
SW388R6
Data Analysis
and Computers I
Slide 24
Solving the problem with SPSS:
Evaluating equality of group variances - 1
The independent-samples t-test assumes that the variances of the
dependent variable for both groups are equal in the population. This
assumption is evaluated with Levene's Test for Equality of Variances.
The null hypothesis for this test states that the variance for both groups
are equal. The desired outcome for this test is to fail to reject the null
hypothesis, which demonstrates equality.
The probability associated with Levene's Test for Equality of Variances
(.161) is greater than alpha (.05), indicating that the 'Equal variances
assumed' formula for the independent samples t-test should be used for
the analysis.
SW388R6
Data Analysis
and Computers I
Slide 25
Solving the problem with SPSS:
Evaluating equality of group variances - 2
Since we failed to reject the hypothesis for Levene’s
test, the 'Equal variances assumed' formula for the
independent samples t-test should be used for the
analysis.
Had the probability associated with Levene’s test
been less than the alpha level, we would have used
the statistics for the ‘Equal variances not assumed’
row in the table.
SW388R6
Data Analysis
and Computers I
Slide 26
Solving the problem with SPSS:
Answering the question - 1
The finding we are trying to verify is:
Survey respondents who were employed by
the federal, state, or local government
completed significantly more years of school
(M = 13.97, SD = 3.27) than survey
respondents who were employed by a private
employer (M = 13.07, SD = 2.84) .
Our first task is to make certain we have
solved the right problem.
First, we check to
make certain we have
the correct groups in
the output.
Second, we verify that
the mean and
standard deviations for
the groups match the
problem statement.
SW388R6
Data Analysis
and Computers I
Slide 27
Solving the problem with SPSS:
Answering the question - 2
The finding we are trying to verify is:
Survey respondents who were employed by the federal, state,
or local government completed significantly more years of
school (M = 13.97, SD = 3.27) than survey respondents who
were employed by a private employer (M = 13.07, SD = 2.84) .
Since the problem states that the mean for one group is
significantly higher than the mean of the other group, the
research hypothesis is a one-tailed test.
We divide the SPSS 2-tailed significance (.080) in half and
make our decision about the null hypothesis by comparing p =
.04 to alpha = .05.
SW388R6
Data Analysis
and Computers I
Slide 28
Solving the problem with SPSS:
Answering the question - 3
The answer to the question is True.
We can include the t-test results in our statement of the
finding:
Survey respondents who were employed by the federal,
state, or local government completed significantly more
years of school (M = 13.97, SD = 3.27) than survey
respondents who were employed by a private employer (M =
13.07, SD = 2.84) , t(255) = 1.761, p = .04 (one-tailed).
SW388R6
Data Analysis
and Computers I
Slide 29
Logic for independent-samples t-test:
Level of measurement
Measurement
level of
independent
variable?
Dichotomous
Interval/ordinal
/nominal
Inappropriate
application of
a statistic
Measurement
level of
dependent
variable?
Interval/ordinal
Strictly speaking, the
test requires an interval
level variable. We will
allow ordinal level
variables with a caution.
Nominal/
Dichotomous
Inappropriate
application of
a statistic
SW388R6
Data Analysis
and Computers I
Slide 30
Logic for independent-samples t-test:
Assumption of normality
Skewness and
Kurtosis
between
-1.0 and +1.0?
Yes
No
Number of cases
in both groups is
at least 30?
Yes
No
Inappropriate
application of
a statistic
SW388R6
Data Analysis
and Computers I
Slide 31
Logic for independent-samples t-test:
Assumption of equality of variances
Yes
Use ‘Equal variances
not assumed’
Probability for Levene
test of equality of
population variances
less than or equal to
alpha?
No
Use ‘Equal variances
assumed’
SW388R6
Data Analysis
and Computers I
Slide 32
Logic for independent-samples t-test:
Means and standard deviations correct
Mean and standard
deviation of both
variables are correct?
Yes
No
False
SW388R6
Data Analysis
and Computers I
Slide 33
Logic for independent-samples t-test:
Decision about null hypothesis
One-tailed or
two-tailed test?
Two-tailed
One-tailed
Divide two-tailed
significance by 2
Add
caution for
ordinal
dependent
variable.
Yes
True
Probability for
t-test less than
or equal to
alpha?
No
False
Power Analysis: Independent-samples T-test
Problem that was False
SW388R6
Data Analysis
and Computers I
Slide 34
This problem uses the data set GSS2000R.Sav to compare the
average score on the variable "number of hours worked in the past
week" [hrs1] for groups of survey respondents defined by the
variable "self-employment" [wrkslf]. Using an independent samples
t-test with an alpha of .05, is the following statement true, true
with caution, false, or an incorrect application of a statistic?
Survey respondents who were self-employed worked significantly
longer hours in the past week (M = 42.04, SD = 13.86) than survey
respondents who were working for someone else (M = 40.55, SD =
12.46) .
1
2
3
4
The answer to this problem was false
because the probability for the t-test was
.29 (one-tailed), greater than the alpha
of 0.05.
True
True with caution
We can conduct a post-hoc power
analysis to determine what number of
False
cases would have been sufficient to have
a better opportunity to find a statistically
Incorrect application of a statistic
significant difference.
SW388R6
Data Analysis
and Computers I
Slide 35
Power Analysis: Statistical Results for
False Independent-samples T-test - 1
The answer to the problem was false
because the one-tailed significance was p =
.29 (.583 ÷ 2), greater than the alpha of
.05.
SW388R6
Data Analysis
and Computers I
Slide 36
Power Analysis: Statistical Results for
False Independent-samples T-test - 2
To calculate the effect size, and
corresponding power, for this problem,
we need a pooled estimate of the
standard deviation for the two groups.
SamplePower will calculate that for us,
we will enter the sample sizes, means,
and standard deviations for the two
groups in SamplePower.
SW388R6
Data Analysis
and Computers I
Access to SPSS’s SamplePower Program
Slide 37
The UT license for SPSS does not
include SamplePower, the SPSS
program for power analysis.
However, the program is available
on the UT timesharing server.
Information about access this
program is available at this site.
SW388R6
Data Analysis
and Computers I
Slide 38
Power Analysis for
Independent-samples T-test - 1
In the SamplePower program on
the ITS Timesharing Systems,
select the New… command from
the File menu.
SW388R6
Data Analysis
and Computers I
Slide 39
Power Analysis for
Independent-samples T-test - 2
First, select the
Means tab to
access the tests for
means.
Second, since we want to enter the
means for our two groups, select the
option button for t-test for 2
(independent) groups with common
variance (Enter means)
Third, click on the
Ok button to enter
the specific values
for our problem.
SW388R6
Data Analysis
and Computers I
Slide 40
Power Analysis for
Independent-samples T-test – 3
I want to my entries to display
two decimal places, instead of
the default of 1, so I click on the
Decimals displayed tool button.
SW388R6
Data Analysis
and Computers I
Slide 41
Power Analysis for
Independent-samples T-test – 4
First, click the up
arrow button on the
spinner for Decimals
for data entry until 2
appears.
Second, click
on the OK
button to close
the dialog box.
SW388R6
Data Analysis
and Computers I
Slide 42
Power Analysis for
Independent-samples T-test - 5
SPSS sets the default test to a twotailed test with an alpha of .05.
Since our test was a one-tailed test
with an alpha of .05, we click on
the text specified as the SPSS
default.
SW388R6
Data Analysis
and Computers I
Slide 43
Power Analysis for
Independent-samples T-test - 6
First, click on the
1 Tailed option on
the Tails panel.
Second, click
on the Ok
button to
change the test
specifications.
SW388R6
Data Analysis
and Computers I
Slide 44
Power Analysis for
Independent-samples T-test - 7
We enter the values from the SPSS
output from the independentsamples t-test for the Population 1
group:
•42.04 for Population Mean
•13.86 for Standard
Deviation
•26 for the N Per Group
Note that SPSS fills in the standard
deviation and N Per Group numbers
for Population 2 with the same
values.
SW388R6
Data Analysis
and Computers I
Slide 45
Power Analysis for
Independent-samples T-test – 8
First, enter the
population mean for the
second group, 40.55.
When we click on the box to change
the Standard Deviation, this message
appears. Since the standard
deviation for our two groups is not
the same, we click on the Yes button.
SW388R6
Data Analysis
and Computers I
Slide 46
Power Analysis for
Independent-samples T-test – 9
We are now able to
enter the standard
deviation for the second
group, 12.46.
SW388R6
Data Analysis
and Computers I
Slide 47
Power Analysis for
Independent-samples T-test – 10
When we click on the box to
change the N Per Group for
the second group, this
message box below
appears.
Since the number of
cases for our two groups
is not the same, we click
on the Yes button.
SW388R6
Data Analysis
and Computers I
Slide 48
Power Analysis for
Independent-samples T-test - 11
We are now able to
enter the N Per
Group for the
second group, 145.
Having entered the
values for the two
groups, we now click on
the Compute button.
SW388R6
Data Analysis
and Computers I
Slide 49
Power Analysis for
Independent-samples T-test - 12
SamplePower tells us
that our power to
obtain statistical
significance was 14%,
translating to a
possible successful
outcome 1 in 7 tries.
SW388R6
Data Analysis
and Computers I
Slide 50
Power Analysis for
Independent-samples T-test – 13
With the mean difference of 1.49
and a pooled standard deviation
of 12.68, we can use a calculator
to compute the effect size of .12
(Cohen’s d), about half of what
would be typically characterized
as a small effect.
Suppose, however, that even
a very small effect of this size
had important consequences.
We can ask ourselves how
large would the sample need
to have been in order to find
a statistically significant
effect.
SW388R6
Data Analysis
and Computers I
Slide 51
Power Analysis for
Independent-samples T-test - 14
To find the group sizes
needed, select Find N
for power of 80% from
the Tools menu.
SW388R6
Data Analysis
and Computers I
Slide 52
Power Analysis for
Independent-samples T-test – 15
This dialog box appears.
SamplePower will need
additional information to
know how it should increase
the size of each group.
Click on the Yes
button to link the
group sample sizes.
SW388R6
Data Analysis
and Computers I
Slide 53
Power Analysis for
Independent-samples T-test - 16
Second, using a
calculator, I compute
that group 2 was
about 6 times larger
than group 1, so I
increase the second
spinner to 6.
First, assuming the
proportion of cases in each
of our groups was
representative of the
population, we mark the
option button to Link
Sample Size in two groups.
Third, click OK to
close the dialog
box.
SW388R6
Data Analysis
and Computers I
Slide 54
Power Analysis for
Independent-samples T-test - 17
To find the the group
sizes needed, again
select Find N for power of
80% from the Tools
menu.
SW388R6
Data Analysis
and Computers I
Slide 55
Power Analysis for
Independent-samples T-test - 18
SamplePower indicates that
we would have needed a total
sample of 3,654 to detect this
very small effect size in the
population.
This very small effect size
would have to have very
important consequences in
order to justify the expense
of collecting samples this
large.