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Chapter 16

The independent samples t -test is a parametric statistical technique used to determine

signifi cant 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 signifi cance 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 = (number of subjects in sample 1+number of subjects in sample 2) −2

Example df = (65 in sample 1+ 67 in sample 2) −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 infl ated

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 signifi cant 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 signifi cance 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 signifi cance of study results. For example,

if a study ’ s α was set at 0.05 and the researcher planned on conducting fi ve t -tests on the

study data, the α would be divided by the fi ve t -tests (0.05 ÷ 5 = 0.01), with a resulting α

of 0.01 to be used to determine signifi cant 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.

EXERCISE

162 EXERCISE 16 • Understanding Independent Samples t-Test

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 withinsamples

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 signifi cance 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 signifi -

cantly 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 signifi cantly lower fear and anxiety

Understanding Independent Samples t-Test • EXERCISE 16 163

TABLE 1 COMPARISON OF GROUPS IN TERMS OF VARIABLES THAT MAY AFFECT PROCEDURAL

PAIN AND ANXIETY LEVELS

Characteristic Buzzy ( n = 88) Control ( n = 88)

χ 2

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)

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)

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)

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.

levels than the control group, according to parents ’ and the observer ’ s reports

Questions to be Graded:

What do degrees of freedom ( df ) mean? Canbulat et al. (2015) did not provide the df s 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.

What are the means and standard deviations ( SD s) 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.
What are the t value and p value for age? What do these results mean?
What are the assumptions for conducting the independent samples t -test?
Are the groups in this study independent or dependent? Provide a rationale for your answer.
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.

Should a Bonferroni procedure be conducted in this study? Provide a rationale for your answer.
What variable has a result of t = − 6.135, p = 0.000? What does the result mean?
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 during IV insertion.

Chapter 17

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 quasiexperimental

design because it is diffi cult 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:

1. The distribution of scores is normal or approximately normal.

2. The dependent variable(s) is(are) measured at interval or ratio levels.

3. Repeated measures data are collected from one group of subjects, resulting in paired

scores.

4. The differences between the paired scores are independent.

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 artifi cial sweetener aspartame in drinks and food, there

are concern and controversy about the mixed research evidence on its neurobehavioral

EXERCISE

172 EXERCISE 17 • Understanding Paired or Dependent Samples t-Test

effects. Thus Lindseth and colleagues (2014) conducted a one-group repeated measures

design to determine the neurobehavioral effects of consuming both low- and highaspartame

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 infl uence 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).

TABLE 2 WITHIN-SUBJECT DIFFERENCES IN NEUROBEHAVIOR SCORES AFTER HIGH

ARelevant 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 signifi cantly

better for participants after their low-aspartame intake period than after their high intake

period ( Table 2 ). Two participants had clinically signifi cant cognitive impairment after

consuming high-aspartame diets. . . . Participants were signifi cantly more depressed after

they consumed the high-aspartame diet compared to when they consumed the lowaspartame

diet ( Table 2 ). . . . Only one participant reported a headache; no difference in

headache incidence between high- and low-aspartame intake periods could be established”

( Lindseth et al., 2014 , p. 190).ND LOW

ASPARTAME INTAKE ( N = 28)

Variable M/ SD/ Paired t -Test /p

Spatial orientation

High-aspartame 14.1/ 4.2/ 2.4/ .03*

Low-aspartame 16.6/ 4.3

Working memory

High-aspartame 730.0/ 152.7/ 1.5/ N.S.

Low-aspartame 761.1/ 201.6

Mood (irritability)

High-aspartame 33.4/ 9.0/ 3.4/ .002**

Low-aspartame 30.5 /7.3

Depression

High-aspartame 36.8 /7.0 /3.8 /.001**

Low-aspartame 34.4/ 6.2

* p < .05.

** p < .01.

M = Mean; SD = Standard deviation; N.S. = Nonsignifi cant.

Lindseth, G. N., Coolahan, S. E., Petros, T. V., & Lindseth, P. D. (2014). Neurobehavioral effects of aspartame consumption.

Chapter 17 Questions to be graded

1. What are the assumptions for conducting a paired or dependent samples t -test in a study? Which

of these assumptions do you think were met by the Lindseth et al. (2014) study?

2. In the introduction, Lindseth et al. (2014) described a “2-week washout between diets.” What

does this mean? Why is this important?

3. What is the paired t -test value for mood (irritability) between the participants ’ consumption of

high- versus low-aspartame diets? Is this result statistically signifi cant? Provide a rationale for

your answer.

4. State the null hypothesis for mood (irritability) that was tested in this study. Was this hypothesis

accepted or rejected? Provide a rationale for your answer.

5. Which t value in Table 2 represents the greatest relative or standardized difference between the

high- and low-aspartame diets? Is this t value statistically signifi cant? Provide a rationale for your

answer.

6. Discuss why the larger t values are more likely to be statistically signifi cant.

7. Discuss the meaning of the results regarding depression for this study. What is the clinical

importance of this result?

8. What is the smallest, paired t -test value in Table 2 ? Why do you think the smaller t values are

not statistically signifi cant?

9. Discuss the clinical importance of these study results about the consumption of aspartame.

Document your answer with a relevant source.

10. Are these study fi ndings related to the consumption of high- and low-aspartame diets ready for

implementation in practice? Provide a rationale for your answer.