The
t-test
is a parametric analysis technique used to determine significant differences between the scores obtained from two groups. The
t-test uses the standard deviation to estimate the standard error of the sampling distribution and examines the differences between the means of the two groups. Since the
t-test is considered fairly easy to calculate, researchers often use it in determining differences between two groups. When interpreting the results of
t-tests, the larger the calculated
t
ratio, in absolute value, the greater the difference between the two groups. The significance of a
t
ratio can be determined by comparison with the critical values in a statistical table for the
t
distribution using the degrees of freedom (df) for the study. The formula for
df
for an independent
t-test is:
df
= number
?
of
?
sobjects
?
in
?
sample
?1 + number
?
of
?
subjects
?
in
?
sample
?2 - 2
The
t-test can only be used once to examine data from two study samples, otherwise the Type 1 error rate (alpha) may be inflated. A Type I error occurs when the researcher rejects the null hypothesis when it is in actuality true. Thus if researchers run multiple
t-tests to evaluate differences of various aspects of a study's data, this is considered a misuse of the
t-test and often leads to an increased risk for a Type I error or finding two groups significantly different when they are not. To correct for the risk of a Type I error, the researcher can perform a Bonferroni procedure. The Bonferroni procedure is a simple calculation in which the alpha is divided by the number of
t-tests run on different aspects of the study data. The resulting number is used as the alpha or level of significance for each of the
t-tests conducted. For example, if a study's alpha was set at 0.05 and the researcher planned on conducting 5
t-tests on the study data, the alpha would be divided by the 5
t-tests (0.05 ÷ 5 = 0.01), with a resulting alpha of 0.01 to be used to determine significant differences in the study. The Bonferroni procedure formula is:
alpha (a) ÷ number
?
of
?t
- tests
?
performed
?
on
?
study
?
data = more
?
stringent
?
study
?a?
to
?
determine
?
the
?
significance
?
of
?
study
?
results .
The
t-test for independent groups includes the following assumptions:
1. The raw scores in the population are normally distributed.
2. The dependent variable(s) is (are) measured at the interval or ratio levels.
3. The two groups examined for differences have equal variance, which is best achieved by a random sample and random assignment to groups.
4. All observations within each group are independent.
The
t-test is robust, meaning the results are reliable even if one of the assumptions has been violated. However, the
t-test is not robust regarding between-samples or within-samples independence assumptions, or with respect to extreme violation of the assumption of normality. Sample groups do not need to be of equal sizes but rather of equal variance. Groups are independent if the two sets of data were not taken from the same subjects and if the scores are not related. Thus, paired or matched groups are dependent, not independent; but a randomly selected sample with random assignment to groups does produce independent groups (Burns & Grove, 2005).
RESEARCH ARTICLE
Source:
Kristofferzon, M., Löfmark, R., & Carlsson, M. (2005). Perceived coping, social support, and quality of life 1 month after myocardial infarction: A comparison between Swedish women and men.
Heart & Lung, 34
(1), 39–50.
Introduction
Kristofferzon, Löfmark, and Carlsson (2005) conducted a comparative-descriptive study to determine if women and men differ in their perceived coping, social support, and quality of life one month post myocardial infarction (MI). The sample of convenience included 171 subjects, 74 women and 97 men. Each participant completed a study-specific questionnaire (demographics and risk factors), the JCS-60 (measured use of coping strategies), the social network and social support questionnaire (measured social participation and emotional support), the SF-36 Health Survey (measured perceived health-related quality of life), and the QLI (measured perceived quality of life). In addition, the researchers conducted a chart review of each participant's medical record. In this study the results showed that “compared with men, women used more evasive and supportive coping and rated psychologic aspects of the heart disease as more problematic to manage. More women perceived available support from friends and grandchildren, and more men perceived available support from their partner. Women rated lower levels in physical and psychologic dimensions of quality of life” (Kristofferzon et al., 2005, p. 39).
Relevant Study Results
“A consecutive series of patients was selected from the medical records in 1 hospital between August 1999 and July 2001 for women and between August 1999 and August 2000 for men. With regard to a lower incidence rate of MI in women, a longer selection period was needed for them…. We decided to include 100 women and 100 men to have a comfortable margin for dropouts.
“An introductory letter, informed consent form, and questionnaires were mailed to eligible subjects 1 month after an acute MI. After 1 week, the first author phoned the patients. Those interested in participating returned the signed consent form and the completed questionnaires to the investigator within 1 to 2 weeks. The same questionnaires were mailed to the subjects on 3 occasions, 1, 4, and 12 months after MI. Data from 1 month are presented in this article” (Kristofferzon et al., 2005, p. 41).
“Of the target population of 338 women, 20% died before inclusion, 35% did not meet the inclusion criteria, and 23% declined participation; of the target population of 317 men, the corresponding numbers were 17%, 27%, and 26%, respectively. … The final sample consisted of 74 women and 97 men” (Kristofferzon et al., 2005, p. 41).
In Table VI, are the quality of life measures reported by Kristofferzon et al. (2005) in their study of women and men following an MI. The level of significance or alpha for this study was set at 0.05.
TABLE VI Quality of Life Experienced by Women and Men (N
= 171)
INSTRUMENTS AND COMPONENTS/SCALES
WOMEN (n
= 74) MEAN (SD)
MEN (n
= 97) MEAN (SD)
t
value (DF
= 169)
P
Value
SF-36 (0 = low QoL, 100 = high QoL)
The Physical Component Score (PCS)*
48.5 (5.7)
51.1 (7.4)
-2.50
0.01
The Mental Component Score (MCS)†
48.2 (7.6)
51.4 (7.5)
-2.74
0.007
Physical Functioning (PF)
51.3 (23.7)
58.6 (24.1)
-1.98
0.049
Role-Physical (RP)
4.7 (14.1)
12.6 (23.7)
-2.54‡
0.007
Bodily Pain (BP)
57.6 (26.2)
62.5 (27.3)
0.24
General Health (GH)
51.1 (17.6)
54.2 (20.1)
0.30
Vitality (VT)
39.8 (19.0)
47.5 (23.2)
-2.31§
0.02
Social Functioning (SF)
61.0 (27.4)
66.1 (23.2)
0.19
Role-Emotional (RE)
27.5 (39.5)
37.8 (42.7)
0.11
Mental Health (MH)
62.3 (22.9)
72.7 (20.1)
-3.15
0.002
QLI (0 = low QoL, 30 = high QoL)
Total Scale
20.1 (3.5)
21.2 (3.6)
-2.06
0.04
Health Functioning
17.9 (4.1)
19.3 (4.6)
-1.99
0.049
Socioeconomic
22.6 (3.6)
22.9 (3.7)
0.58
Psychologic/spiritual
19.6 (4.6)
21.1 (4.3)
-2.10
0.04
Family (N
= 69 women and 94 men)
25.6 (4.7)
26.0 (3.9)
0.51
Kristofferzon, M., Löfmark, R., & Carlsson, M. (2005). Perceived coping, social support, and quality of life 1 month after myocardial infarction: A comparison between Swedish women and men.
Heart & Lung, 34(1), p. 47.
* PCS = PF, RP, BP, GH.
_ MCS = VT, SF, RE, MH.
_
df
= 161.
_
df
= 168.
QoL
= Quality of life;
QLI, Quality-of-Life Index-Cardiac Version.
STUDY QUESTIONS
1.
t
= -1.99 describes the difference between women and men post myocardial infarction (MI) for what variable?
2. Consider
t
= -2.74 and
t
= -2.31. Which calculated
t
ratio has the smaller
p
value? Provide a rationale for your answer.
3. Examine the results in Table VI. Which
t
ratio listed in the table had the largest
p
value? What was the focus of this
t-test, and were the results significant? Provide a rationale for your answer.
4. What is
df?
Why is it important to know the
df
for a
t
ratio? How would you calculate the
df
for a
t-test, and what is the
df
for this study?
5. What is the cause of an increased risk for Type I errors when
t-tests are conducted? How might researchers eliminate the increased risk for a Type I error in a study?
6. Given the information presented in Table VI, calculate a Bonferroni procedure for this study.
7. Does this study meet the assumptions for the
t-test? Provide a rationale for your answer.
8. What sampling method did the researchers use in this study? Provide a rationale for your answer.
9. What level of data is analyzed by means and standard deviations? Is this level of data compatible with the assumptions for the
t-test? Provide a rationale for your answer.
10. Is the sample size adequate to detect significant differences between the two groups in this study?
ANSWERS TO STUDY QUESTIONS
1.
t
= -1.99 describes the difference in health functioning between women and men after a MI.
2.
t
= -2.74,
p
= 0.007;
t
= -2.31,
p
= 0.02.
t
= -2.74 has the smaller
p
value at
p
= 0.007 than
t
= -2.31 with
p
= 0.02. The smaller
p
value indicates more significant findings.
3. Both
t
= -1.98 (physical functioning) and
t
= -1.99 (health functioning) had equal and the largest
p
values at
p
= 0.049 that were still considered statistically significant. These two
t
ratios indicated the differences between males and females for physical functioning and health functioning in this sample. These
t
values are significant because the
p
= 0.049 value is smaller than alpha (a) that was set at 0.05 for this study.
4.
df
= degrees of freedom. Degrees of freedom (df) is a mathematical equation that describes the freedom of a particular score's value to vary based on the other existing scores’ values and the sum of the scores (Burns & Grove, 2005). The
df
for an analysis technique allows you to look up
t
ratios on a statistical table that includes
t
distributions to determine their significance. The
df
calculations vary based on the analysis technique conducted. Thus, the formula for
df
for the
t-test for independent groups is:
5. The conduct of multiple
t-tests causes an increased risk for Type I errors. If only one
t-test is performed on study data, the risk of Type I error does not increase. A Bonferroni procedure reduces the risk for Type I errors.
6. The Bonferroni procedure is calculated by alpha ÷ number of
t-tests conducted on study data. For this study only 9
t
values are provided in Table VI, but the
p
values indicate that 15
t-tests were conducted and only 9 of these were significant. Thus, the Bonferroni calculation includes:
0.05( alpha ) ÷ 15( number
?
of
?t
- tests
?
conducted
?
in
?
a
?
study ) = 0.0033( Bonferroni
?
result )
7. Answers may vary. Yes, the study meets the required assumptions for the
t-test. The researchers do not indicate if the scores from the two groups are normally distributed, so it is assumed that they are. The study variables are measured at least at the interval level as indicated by the measurement methods used in the study. The researchers mentioned the longer length of time required to recruit an adequate number of women to ensure a more equal variance between the groups. However, the variance of scores would have been ensured more if the original sample had been randomly selected. The two groups were independent since they were formed based on gender (male and female) with no intent to match subjects on any variable.
8. A nonprobability, convenience sampling method was used in the study. The researchers recruited consecutive patients from one hospital over a period of time, which is consistent with a convenience sampling method. If a random sampling method had been used to obtain the sample, the researcher would have indicated this in the sample section of the study.
9. Means and standard deviations are calculated for variables that are measured at the interval and ratio levels. These analysis techniques are used to describe study variables. The
t-test is a parametric analysis technique to detect differences between two groups. The dependent variables in a study must be measured at the interval or ratio levels in order for a
t-test to be conducted (see the assumptions for the
t-test).
10. Answers may vary. The sample size is adequate since 9
t
values were significant in this study. If significant differences are detected, then the sample size is adequate. However, if you use the Bonferroni procedure, there is only one
t
value (t
= -3.15,
p
= 0.002) that is significant at the 0.0033 level of significance identified in Question 6. Thus, the sample size might be viewed as too small since only one
t
value is significant. Or one might conclude that the differences between the males and females post MI are not significant.
Name:____________________________________________ Class: ____________________
Date: _________________________________________________________________________________
_ EXERCISE 29 Questions to be Graded
1. Were the groups in this study independent or dependent? Provide a rationale for your answer.
2.
t
= -3.15 describes the difference between women and men for what variable in this study? Is this value significant? Provide a rationale for your answer.
3. Is
t
= -1.99 significant? Provide a rationale for your answer. Discuss the meaning of this result in this study.
4. Examine the
t
ratios in Table VI. Which
t
ratio indicates the largest difference between the males and females post MI in this study? Is this
t
ratio significant? Provide a rationale for your answer.
5. Consider
t
= -2.50 and
t
= -2.54. Which
t
ratio has the smaller
p
value? Provide a rationale for your answer. What does this result mean?
6. What is a Type I error? Is there a risk of a Type I error in this study? Provide a rationale for your answer.
7. Should a Bonferroni procedure be conducted in this study? Provide a rationale for your answer.
8. If researchers conducted 9
t-tests on their study data. What alpha level should be used to determine significant differences between the two groups in the study? Provide your calculations.
9. The authors reported multiple
df
values in Table VI. Why were different
df
values reported for this study?
10. What does the
t
value for the Physical Component Score tell you about men and women post MI? If this result was consistent with previous research, how might you use this knowledge in your practice?
(Grove 217)
Grove, Susan K..
Statistics for Health Care Research: A Practical Workbook. W.B. Saunders Company, 022007. .