Post a brief outline of the case study and consider the conclusion that the vocational rehabilitation intervention program may be effective at promoting full-time employment. What statistical information shows whether the program was effective (or not)? Review the factors that limit the internal validity of a study (history, maturation, testing, instrumentation, statistical regression, selection bias, and attrition).Select and explain which of these factors might limit the ability to draw conclusions regarding cause-and-effect relationships.
A Short Course in Statistics
This information was prepared to call your attention to some basic concepts underlying
statistical procedures and to illustrate what types of research questions can be
addressed by different statistical tests. You may not fully understand these tests without
further study. However, you are strongly encouraged to note distinctions related to the
type of measurement used in gathering data and the choice of statistical tests. Feel free
to post questions in the Contact the Instructor section of the course.
Statistical Symbols
µ mu (population mean)
? alpha (degree of error acceptable for incorrectly rejecting the null hypothesis,
probability that results are unlikely to occur by chance)
? (not equal)
? (greater than or equal to)
? less than or equal to)
? (sample correlation)
? rho (population correlation)
t r (t score)
z (standard score based on standard deviation)
?2 Chi-square (statistical test for variables that are not interval or ratio scale [i.e.,
nominal or ordinal])
p (probability that results are due to chance)
Descriptives
Descriptives are statistical tests that summarize a data set.
They include calculations of measures of central tendency (mean, median, and mode)
and dispersion (e.g., standard deviation and range).
Note: The measures of central tendency depend on the measurement level of the
variable (nominal, ordinal, interval, or ratio). If you do not recall the definitions for these
levels of measurement, see https://www.questionpro.com/blog/nominal-ordinal-intervalratio/
You can only calculate a mean and standard deviation for interval or ratio scale
variables.
For nominal or ordinal variables, you can examine the frequency of responses. For
example, you can calculate the percentage of participants who are male and female; or
the percentage of survey respondents who are in favor, against, or undecided.
Often nominal data is recorded with numbers, e.g., male=1, female=2. Sometimes
people are tempted to calculate a mean using these coding numbers. But that would be
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meaningless. Many questionnaires (even course evaluations) use a Likert scale to
represent attitudes along a continuum (e.g., Strongly like
Strongly dislike). These too
are often assigned a number for data entry, e.g., 15. Suppose that most of the
responses were in the middle of a scale (3 on a scale of 15). A researcher could
observe that the mode is 3, but it would not be reasonable to say that the average
(mean) is 3 unless there were exact differences between 1 and 2, 2 and 3, etc. The
numbers on a scale such as this are ordered from low to high or high to low, but there is
no way to say that there is a quantifiably equal difference between each of the choices.
In other words, the responses are ordered but not necessarily equal. Strongly agree is
not five times as large as strongly disagree. (See the textbook for differences between
ordinal and interval scale measures.)
Inferential Statistics
Statistical tests for analysis of differences or relationships are inferential,
allowing a researcher to infer relationships between variables.
All statistical tests have what are called assumptions. These are essentially rules that
indicate that the analysis is appropriate for the type of data. Two key types of
assumptions relate to whether the samples are random and the measurement levels.
Other assumptions have to do with whether the variables are normally distributed. The
determination of statistical significance is based on the assumption of the normal
distribution. A full course in statistics would be needed to explain this fully. The key point
for our purposes is that some statistical procedures require a normal distribution and
others do not.
Understanding Statistical Significance
Regardless of what statistical test you use to test hypotheses, you will be looking to see
whether the results are statistically significant. The statistic p is the probability that the
results of a study would occur simply by chance. Essentially, a p that is less than or
equal to a predetermined (?) alpha level (commonly .05) means that we can reject a null
hypothesis. A null hypothesis always states that there is no difference or no relationship
between the groups or variables. When we reject the null hypothesis, we conclude (but
dont prove) that there is a difference or a relationship. This is what we generally want to
know.
Parametric Tests
Parametric tests are tests that require variables to be measured at interval or ratio
scale and for the variables to be normally distributed.
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These tests compare the means between groups, which is why they require the data to
be at an interval or ratio scale. They make use of the standard deviation to determine
whether the results are likely to occur or very unlikely in a normal distribution. If they are
very unlikely to occur, then they are considered statistically significant. This means that
the results are unlikely to occur simply by chance.
The T-Test
Common uses:
To compare mean from a sample group to a known mean from a population
To compare the mean between two samples
o The research question for a t-test comparing the mean scores between
two samples is: Is there a difference in scores between group 1 and group
2? The hypotheses tested would be:
H0: µgroup1 = µgroup2
H1: µgroup1 ? µgroup2
To compare pre- and post-test scores for one sample
o The research question for a t-test comparing the mean scores for a
sample with pre and posttests is: Is there a difference in scores between
time 1 and time 2? The hypotheses tested would be :
H0: µpre = µpost
H1: µpre ? µpost
Example of the form for reporting results: The results of the test were not statistically
significant, t (57) = .282, p = .779, thus the null hypothesis is not rejected. There is not a
difference in between pre and post scores for participants in terms of a measure of
knowledge (for example).
An explanation: The t is a value calculated using means and standard deviations and a
relationship to a normal distribution. If you calculated the t using a formula, you would
compare the obtained t to a table of t values that is based on one less than the number
of participants (n-1). n-1 represents the degrees of freedom. The obtained t must be
greater than a critical value of t in order to be significant. For example, if statistical
analysis software calculated that p = .779, this result is much greater than .05, the usual
alpha-level which most researchers use to establish significance. In order for the t-test
to be significant, it would need to have a p ? .05.
ANOVA (Analysis of Variance)
Common uses: Similar to the t-test. However, it can be used when there are more than
two groups.
The hypotheses would be
H0: µgroup1 = µgroup2 = µgroup3 = µgroup4
H1: The means are not all equal (some may be equal)
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Correlation
Common use: to examine whether two variables are related, that is, they vary together.
The calculation of a correlation coefficient (r or rho) is based on means and standard
deviations. This requires that both (or all) variables are measured at an interval or ratio
level.
The coefficient can range from -1 to +1. An r of 1 is a perfect correlation. A + means that
as one variable increases, so does the other. A means that as one variable increases,
the other decreases.
The research question for correlation is: Is there a relationship between variable 1 and
one or more other variables?
The hypotheses for a Pearson correlation:
H0: ? = 0 (there is no correlation)
H1: ? ? 0 (there is a real correlation)
Nonparametric Tests
Nonparametric tests are tests that do not require variables to be measured at
interval or ratio scale and do not require the variables to be normally distributed.
Chi-Square
Common uses: Chi-square tests of independence and measures of association and
agreement for nominal and ordinal data.
The research question for a chi-square test for independence is: Is there a relationship
between the independent variable and a dependent variable?
The hypotheses are:
H0 (The null hypothesis) There is no difference in the proportions in each category of
one variable between the groups (defined as categories of another variable).
Or:
The frequency distribution for variable 2 has the same proportions for both categories of
variable 1.
H1 (The alternative hypothesis) There is a difference in the proportions in each category
of one variable between the groups (defined as categories of another variable).
The calculations are based on comparing the observed frequency in each category to
what would be expected if the proportions were equal. (If the proportions between
observed and expected frequencies are equal, then there is no difference.)
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Other Nonparametric Tests
Spearman rho: A correlation test for rank ordered (ordinal scale) variables.
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1
Chi-Square
SOCW 6311
Molly, an administrator with a regional organization that advocates for alternatives to
long-term prison sentences for nonviolent offenders, asked a team of researchers to
conduct an outcome evaluation of a new vocational rehabilitation program for recently
paroled prison inmates. The primary goal of the program is to promote full-time
employment among its participants.
To evaluate the program, the evaluators decided to use a quasi-experimental research
design. The program enrolled 30 individuals to participate in the new program.
Additionally, there was a waiting list of 30 other participants who planned to enroll after
the first group completed the program. After the first group of 30 participants completed
the vocational program (the intervention group), the researchers compared those
participants levels of employment with the 30 on the waiting list (the comparison
group).
The research question for the study is: Is there a relationship between the independent
variable, treatment, and the dependent variable, employment level? In other words, is
there a difference in the number of participants who are not employed, employed parttime, and employed full-time in the program and the control group (i.e., waitlist group)?
Data Collection and Analysis
In order to collect data on employment levels, the probation officers for each of the 60
people in the sample (those in both the intervention and comparison groups) completed
a short survey on the status of each client in the sample. The survey contained
demographic questions that included an item that inquired about the employment level
of the client. This was measured through variables identified as none, part-time, or fulltime. A hard copy of the survey was mailed to each probation officer and a stamped,
self-addressed envelope was provided for return of the survey to the researchers.
After the surveys were returned, the researchers entered the data into an SPSS
program for statistical analysis. Because both the independent variable (participation in
the vocational rehabilitation program) and dependent variable (employment outcome)
used nominal/categorical measurement, the bivariate statistic selected to compare the
outcome of the two groups was the Pearson chi-square.
The chi-square test for independence is used to determine whether there is a
relationship between the two variables that are categorical in the level of measurement.
In this case, the variables are: employment level and treatment condition. It tests
whether there is a difference between groups.
The hypotheses are:
H0 (The null hypothesis): There is no difference in the proportions of individuals in the
three employment categories between the treatment group and the waitlist group. In
© 2022 Walden University, LLC. Adapted from Plummer, S.-B., Makris, S., & Brocksen, S. M. (Eds.). (2014). Social
work case studies: Concentration year. Laureate International Universities Publishing.
2
other words, the frequency distribution for variable 2 (employment) has the same
proportions for both categories of variable 1 (program participation).
Note: It is the null hypothesis that is actually tested by the statistic. A chi-square
statistic that is found to be statistically significant (p < .05) indicates that we can
reject the null hypothesis (understanding that there is less than a 5% chance that
the relationship between the variables is due to chance).
H1 (The alternative hypothesis): There is a difference in the proportions of individuals in
the three employment categories between the treatment group and the waitlist group.
Note: The alternative hypothesis states that there is a difference. It would allow
us to say that it appears that the treatment (vocational rehab program) is
effective in increasing the employment status of participants.
Results
After all of the information was entered into the SPSS program, the following output
charts were generated:
Table 1. Case Processing Summary
Program
Participation
*Employment
N
59
Valid
Percent
98.3%
Cases
Missing
N
Percent
1
1.7%
N
60
Total
Percent
100.0%
The first table, titled Case Processing Summary, provided the sample size (N = 59).
Information for one of the 60 participants was not available, while the information was
collected for all of the other 59 participants.
Table 2. Program Participation
*Employment Cross Tabulation
5
16.7%
Employment
Part-Time Full-Time
7
18
23.3%
60.0%
30
100.0%
16
55.2%
7
24.1%
6
20.7%
29
100.0%
21
35.6%
14
23.7%
24
40.7%
59
100.0%
None
Program
Intervention
Participation Group
Total
Count %
within
Program
Participation
Comparison Count %
Group
within
Program
Participation
Count %
within
Total
© 2022 Walden University, LLC. Adapted from Plummer, S.-B., Makris, S., & Brocksen, S. M. (Eds.). (2014). Social
work case studies: Concentration year. Laureate International Universities Publishing.
3
Program
Participation
The second table, Program Participation Employment Cross Tabulation, provided the
frequency table, which showed that among participants in the intervention group, 18 or
60% were found to be employed full time, while 7 or 23% were found to be employed
part time, and 5 or 17% were unemployed. The corresponding numbers for the
comparison group (parolees who had not yet enrolled in the program but were on the
waiting list for admission) showed that only 6 or 21% were employed full-time, while 7 or
24% were employed part time, and 16 or 55% were unemployed.
Table 3. Chi-Square Tests
Value
df
Asymp. Sig
(2-sided)
.003
.002
.001
Pearson Chi-Square
11.748a
2
Likelihood Ratio
12.321
2
Linear-by-Linear
11.548
1
Association
N of Valid Cases
59
a.
0 cells (.0%) have expected count less than 5. The minimum expected count is 6.88.
The third table, which provided the outcome of the Pearson chi-square test, found that
the difference between the intervention and comparison groups were highly significant,
with a p value of .003, which is significantly beyond the usual alpha-level of .05 that
most researchers use to establish significance.
Discussion
These results indicate that the vocational rehabilitation intervention program may be
effective at promoting full-time employment among recently paroled inmates. However,
there are multiple limitations to this study, including that 1) no random assignment was
used, and 2) it is possible that differences between the groups were due to preexisting
differences among the participants (such as selection bias).
Potential future studies could include a matched comparison group or, if possible, a
control group. In addition, future studies should assess not only whether or not a
recently paroled individual obtains employment but also the degree to which he or she
is able to maintain employment, earn a living wage, and satisfy other conditions of
probation.
© 2022 Walden University, LLC. Adapted from Plummer, S.-B., Makris, S., & Brocksen, S. M. (Eds.). (2014). Social
work case studies: Concentration year. Laureate International Universities Publishing.
