Understanding Independent Samples t-Test
Statistical Technique in Review
The independent samples t-test is a parametric statistical technique used to determine significant differences between the scores obtained from two samples or groups. Since the t-test is considered fairly easy to calculate, researchers often use it in determining differences between two groups. The t-test examines the differences between the means of the two groups in a study and adjusts that difference for the variability (computed by the standard error) among the data. 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 (see Appendix A Critical Values for Student’s t Distribution at the back of this text). The formula for df for an independent t-test is as follows:
df=(numberofsubjectsinsample1+numberofsubjectsinsample2)−2
Exampledf=(65insample1+67insample2)−2=132−2=130
The t-test should be conducted only once to examine differences between two groups in a study, because conducting multiple t-tests on study data can result in an inflated Type 1 error rate. A Type I error occurs when the researcher rejects the null hypothesis when it is in actuality true. Researchers need to consider other statistical analysis options for their study data rather than conducting multiple t-tests. However, if multiple t-tests are conducted, researchers can perform a Bonferroni procedure or more conservative post hoc tests like Tukey’s honestly significant difference (HSD), Student-Newman-Keuls, or Scheffé test to reduce the risk of a Type I error. Only the Bonferroni procedure is covered in this text; details about the other, more stringent post hoc tests can be found in Plichta and Kelvin (2013) and Zar (2010).
The Bonferroni procedure is a simple calculation in which the alpha is divided by the number of t-tests conducted 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. The Bonferroni procedure formula is as follows: alpha (α) ÷ number of t-tests performed on study data = more stringent study α to determine the significance of study results. For example, if a study’s α was set at 0.05 and the researcher planned on conducting five t-tests on the study data, the α would be divided by the five t-tests (0.05 ÷ 5 = 0.01), with a resulting α of 0.01 to be used to determine significant differences in the study.
The t-test for independent samples or 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 scores or observations collected within each group are independent or not related to other study scores or observations.
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. 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 (Grove, Burns, & Gray, 2013; Plichta & Kelvin, 2013). This exercise focuses on interpreting and critically appraising the t-tests results presented in research reports. Exercise 31 provides a step-by-step process for calculating the independent samples t-test.
Research Article
Source
Canbulat, N., Ayhan, F., & Inal, S. (2015). Effectiveness of external cold and vibration for procedural pain relief during peripheral intravenous cannulation in pediatric patients. Pain Management Nursing, 16(1), 33–39.
Introduction
Canbulat and colleagues (2015, p. 33) conducted an experimental study to determine the “effects of external cold and vibration stimulation via Buzzy on the pain and anxiety levels of children during peripheral intravenous (IV) cannulation.” Buzzy is an 8 × 5 × 2.5 cm battery-operated device for delivering external cold and vibration, which resembles a bee in shape and coloring and has a smiling face. A total of 176 children between the ages of 7 and 12 years who had never had an IV insertion before were recruited and randomly assigned into the equally sized intervention and control groups. During IV insertion, “the control group received no treatment. The intervention group received external cold and vibration stimulation via Buzzy . . . Buzzy was administered about 5 cm above the application area just before the procedure, and the vibration continued until the end of the procedure” (Canbulat et al., 2015, p. 36). Canbulat et al. (2015, pp. 37–38) concluded that “the application of external cold and vibration stimulation were effective in relieving pain and anxiety in children during peripheral IV” insertion and were “quick-acting and effective nonpharmacological measures for pain reduction.” The researchers concluded that the Buzzy intervention is inexpensive and can be easily implemented in clinical practice with a pediatric population.
Relevant Study Results
The level of significance for this study was set at α = 0.05. “There were no differences between the two groups in terms of age, sex [gender], BMI, and preprocedural anxiety according to the self, the parents’, and the observer’s reports (p > 0.05) (Table 1). When the pain and anxiety levels were compared with an independent samples t test, . . . the children in the external cold and vibration stimulation [intervention] group had significantly lower pain levels than the control group according to their self-reports (both WBFC [Wong Baker Faces Scale] and VAS [visual analog scale] scores; p < 0.001) (Table 2). The external cold and vibration stimulation group had significantly lower fear and anxiety 163levels than the control group, according to parents’ and the observer’s reports (p < 0.001) (Table 3)” (Canbulat et al., 2015, p. 36).
TABLE 1
COMPARISON OF GROUPS IN TERMS OF VARIABLES THAT MAY AFFECT PROCEDURAL PAIN AND ANXIETY LEVELS
| Characteristic | Buzzy (n = 88) | Control (n = 88) | χ2 p |
| Sex | |||
| Female (%), n | 11 (12.5) | 13 (14.8) | .82 |
| Male (%), n | 77 (87.5) | 75 (85.2) | .41 |
| Characteristic | Buzzy (n = 88) | Control (n = 88) | t p |
| Age (mean ± SD) | 8.25 ± 1.51 | 8.61 ± 1.69 | −1.498 .136 |
| BMI (mean ± SD) | 25.41 ± 6.74 | 26.94 ± 8.68 | −1.309 .192 |
| Preprocedural anxiety | |||
| Self-report (mean ± SD) | 2.03 ± 1.29 | 2.11 ± 1.58 | −0.364 .716 |
| Parent report (mean ± SD) | 2.11 ± 1.20 | 2.17 ± 1.42 | −0.285 .776 |
| Observer report (mean ± SD) | 2.18 ± 1.17 | 2.24 ± 1.37 | −0.295 .768 |
BMI, body mass index.
Canbulat, N., Ayban, F., & Inal, S. (2015). Effectiveness of external cold and vibration for procedural pain relief during peripheral intravenous cannulation in pediatric patients. Pain Management Nursing, 16(1), p. 36.
TABLE 2
COMPARISON OF GROUPS’ PROCEDURAL PAIN LEVELS DURING PERIPHERAL IV CANNULATION
| Buzzy (n = 88) | Control (n = 88) | t p |
|
| Procedural self-reported pain with WBFS (mean ± SD) | 2.75 ± 2.68 | 5.70 ± 3.31 | −6.498 0.000 |
| Procedural self-reported pain with VAS (mean ± SD) | 1.66 ± 1.95 | 4.09 ± 3.21 | −6.065 0.000 |
IV, intravenous; WBFS, Wong-Baker Faces Scale; SD, standard deviation; VAS, visual analog scale.
Canbulat, N., Ayban, F., & Inal, S. (2015). Effectiveness of external cold and vibration for procedural pain relief during peripheral intravenous cannulation in pediatric patients. Pain Management Nursing, 16(1), p. 37.
TABLE 3
COMPARISON OF GROUPS’ PROCEDURAL ANXIETY LEVELS DURING PERIPHERAL IV CANNULATION
| Procedural Child Anxiety | Buzzy (n = 88) | Control (n = 88) | t p |
| Parent reported (mean ± SD) | 0.94 ± 1.06 | 2.09 ± 1.39 | −6.135 0.000 |
| Observer reported (mean ± SD) | 0.92 ± 1.03 | 2.14 ± 1.34 | −6.745 0.000 |
SD, standard deviation; IV, intravenous.
Canbulat, N., Ayban, F., & Inal, S. (2015). Effectiveness of external cold and vibration for procedural pain relief during peripheral intravenous cannulation in pediatric patients. Pain Management Nursing, 16(1), p. 37.
Study Questions
1. What type of statistical test was conducted by Canbulat et al. (2015) to examine group differences in the dependent variables of procedural pain and anxiety levels in this study? What two groups were analyzed for differences?
2. What did Canbulat et al. (2015) set the level of significance, or alpha (α), at for this study?
3. What are the t and p (probability) values for procedural self-reported pain measured with a visual analog scale (VAS)? What do these results mean?
4. What is the null hypothesis for observer-reported procedural anxiety for the two groups? Was this null hypothesis accepted or rejected in this study? Provide a rationale for your answer.
5. What is the t-test result for BMI? Is this result statistically significant? Provide a rationale for your answer. What does this result mean for the study?
6. What causes an increased risk for Type I errors when t-tests are conducted in a study? How might researchers reduce the increased risk for a Type I error in a study?
7. Assuming that the t-tests presented in Table 2 and Table 3 are all the t-tests performed by Canbulat et al. (2015) to analyze the dependent variables’ data, calculate a Bonferroni procedure for this study.
8. Would the t-test for observer-reported procedural anxiety be significant based on the more stringent α calculated using the Bonferroni procedure in question 7? Provide a rationale for your answer.
9. The results in Table 1 indicate that the Buzzy intervention group and the control group were not significantly different for gender, age, body mass index (BMI), or preprocedural anxiety (as measured by self-report, parent report, or observer report). What do these results indicate about the equivalence of the intervention and control groups at the beginning of the study? Why are these results important?
10. Canbulat et al. (2015) conducted the χ2 test to analyze the difference in sex or gender between the Buzzy intervention group and the control group. Would an independent samples t-test be appropriate to analyze the gender data in this study (review algorithm in Exercise 12)? Provide a rationale for your answer.
Answers to Study Questions
1. An independent samples t-test was conducted to examine group differences in the dependent variables in this study. The two groups analyzed for differences were the Buzzy experimental or intervention group and the control group.
2. The level of significance or alpha (α) was set at 0.05.
3. The result was t = −6.065, p = 0.000 for procedural self-reported pain with the VAS (see Table 2). The t value is statistically significant as indicated by the p = 0.000, which is less than α = 0.05 set for this study. The t result means there is a significant difference between the Buzzy intervention group and the control group in terms of the procedural self-reported pain measured with the VAS. As a point of clarification, p values are never zero in a study. There is always some chance of error.
4. The null hypothesis is: There is no difference in observer-reported procedural anxiety levels between the Buzzy intervention and the control groups for school-age children. The t = −6.745 for observer-reported procedural anxiety levels, p = 0.000, which is less than α = 0.05 set for this study. Since this study result was statistically significant, the null hypothesis was rejected.
5. The t = −1.309 for BMI. The nonsignificant p = .192 for BMI is greater than α = 0.05 set for this study. The nonsignificant result means there is no statistically significant difference between the Buzzy intervention and control groups for BMI. The two groups need to be similar for demographic variables to decrease the potential for error and increase the likelihood that the results are an accurate reflection of reality.
6. The conduct of multiple t-tests causes an increased risk for Type I errors. If only one t-test is conducted on study data, the risk of Type I error does not increase. The Bonferroni procedure and the more stringent Tukey’s honestly significant difference (HSD), Student Newman-Keuls, or Scheffé test can be calculated to reduce the risk of a Type I error (Plichta & Kelvin, 2013; Zar, 2010).
7. The Bonferroni procedure is calculated by alpha ÷ number of t-tests conducted on study variables’ data. Note that researchers do not always report all t-tests conducted, especially if they were not statistically significant. The t-tests conducted on demographic data are not of concern. Canbulat et al. reported the results of four t-tests conducted to examine differences between the intervention and control groups for the dependent variables procedural self-reported pain with WBFS, procedural self-reported pain with VAS, parent-reported anxiety levels, and observer-reported anxiety levels. The Bonferroni calculation for this study: 0.05 (alpha) ÷ number of t-tests conducted = 0.05 ÷ 4 = 0.0125. The new α set for the study is 0.0125.
8. Based on the Bonferroni result = 0.0125 obtained in Question 7, the t = −6.745, p = 0.000, is still significant since it is less than 0.0125.
9. The intervention and control groups were examined for differences related to the demographic variables gender, age, and BMI and the dependent variable preprocedural anxiety that might have affected the procedural pain and anxiety posttest levels in the children 7 to 12 years old. These nonsignificant results indicate the intervention and control groups were similar or equivalent for these variables at the beginning of the study. Thus, Canbulat et al. (2015) can conclude the significant differences found between the two groups for procedural pain and anxiety levels were probably due to the effects of the intervention rather than sampling error or initial group differences.
10. No, the independent samples t-test would not have been appropriate to analyze the differences in gender between the Buzzy intervention and control groups. The demographic variable gender is measured at the nominal level or categories of females and males. Thus, the χ2 test is the appropriate statistic for analyzing gender data (see Exercise 19). In contrast, the t-test is appropriate for analyzing data for the demographic variables age and BMI measured at the ratio level.
EXERCISE 16 Questions to Be Graded
Follow your instructor’s directions to submit your answers to the following questions for grading. Your instructor may ask you to write your answers below and submit them as a hard copy for grading. Alternatively, your instructor may ask you to use the space below for notes and submit your answers online at http://evolve.elsevier.com/Grove/Statistics/ under “Questions to Be Graded.”
Name: _______________________________________________________ Class: _____________________
Date: ___________________________________________________________________________________
1. What do degrees of freedom (df) mean? Canbulat et al. (2015) did not provide the dfs in their study. Why is it important to know the df for a t ratio? Using the df formula, calculate the df for this study.
2. What are the means and standard deviations (SDs) for age for the Buzzy intervention and control groups? What statistical analysis is conducted to determine the difference in means for age for the two groups? Was this an appropriate analysis technique? Provide a rationale for your answer.
3. What are the t value and p value for age? What do these results mean?
4. What are the assumptions for conducting the independent samples t-test?
5. Are the groups in this study independent or dependent? Provide a rationale for your answer.
6. What is the null hypothesis for procedural self-reported pain measured with the Wong Baker Faces Scale (WBFS) for the two groups? Was this null hypothesis accepted or rejected 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. What variable has a result of t = −6.135, p = 0.000? What does the result mean?
9. In your opinion, is it an expected or unexpected finding that both t values on Table 2 were found to be statistically significant. Provide a rationale for your answer.
10. Describe one potential clinical benefit for pediatric patients to receive the Buzzy intervention that combined cold and vibration
Understanding Paired or Dependent Samples t-Test
Statistical Technique in Review
The paired or dependent samples t-test is a parametric statistical procedure calculated to determine differences between two sets of repeated measures data from one group of people. The scores used in the analysis might be obtained from the same subjects under different conditions, such as the one group pretest–posttest design. With this type of design, a single group of subjects experiences the pretest, treatment, and posttest. Subjects are referred to as serving as their own control during the pretest, which is then compared with the posttest scores following the treatment. Paired scores also result from a one-group repeated measures design, where one group of participants is exposed to different levels of an intervention. For example, one group of participants might be exposed to two different doses of a medication and the outcomes for each participant for each dose of medication are measured, resulting in paired scores. The one group design is considered a weak quasi-experimental design because it is difficult to determine the effects of a treatment without a comparison to a separate control group (Shadish, Cook, & Campbell, 2002).
A less common type of paired groups is when the groups are matched as part of the design to ensure similarities between the two groups and thus reduce the effect of extraneous variables (Grove, Burns, & Gray, 2013; Shadish et al., 2002). For example, two groups might be matched on demographic variables such as gender, age, and severity of illness to reduce the extraneous effects of these variables on the study results. The assumptions for the paired samples t-test are as follows:
Research Article
Source
Lindseth, G. N., Coolahan, S. E., Petros, T. V., & Lindseth, P. D. (2014). Neurobehavioral effects of aspartame consumption. Research in Nursing & Health, 37(3), 185–193.
Introduction
Despite the widespread use of the artificial sweetener aspartame in drinks and food, there are concern and controversy about the mixed research evidence on its neurobehavioral 172effects. Thus Lindseth and colleagues (2014) conducted a one-group repeated measures design to determine the neurobehavioral effects of consuming both low- and high-aspartame diets in a sample of 28 college students. “The participants served as their own controls. . . . A random assignment of the diets was used to avoid an error of variance for possible systematic effects of order” (Lindseth et al., 2014, p. 187). “Healthy adults who consumed a study-prepared high-aspartame diet (25 mg/kg body weight/day) for 8 days and a low-aspartame diet (10 mg/kg body weight/day) for 8 days, with a 2-week washout between the diets, were examined for within-subject differences in cognition, depression, mood, and headache. Measures included weight of foods consumed containing aspartame, mood and depression scales, and cognitive tests for working memory and spatial orientation. When consuming high-aspartame diets, participants had more irritable mood, exhibited more depression, and performed worse on spatial orientation tests. Aspartame consumption did not influence working memory. Given that the higher intake level tested here was well below the maximum acceptable daily intake level of 40–50 mg/kg body weight/day, careful consideration is warranted when consuming food products that may affect neurobehavioral health” (Lindseth et al., 2014, p. 185).
Relevant Study Results
“The mean age of the study participants was 20.8 years (SD = 2.5). The average number of years of education was 13.4 (SD = 1.0), and the mean body mass index was 24.1 (SD = 3.5). . . . Based on Vandenberg MRT scores, spatial orientation scores were significantly better for participants after their low-aspartame intake period than after their high intake period (