COMPARING TWO INDEPENDENT GROUPS 2
There are several assumptions sizes for the use of an independent samples t tests. State each
of these and the implications should these assumptions be violated. Is it possible for a p value
to equal 0? Why or why not?
Sometimes before a researcher can perform a test, they have to ensure the validity of the
statistical significances drawn from the test. Meeting the assumptions of statistical tests are often
an overlooked and forgotten step to reporting valid results (Patino & Ferreria, 2018). The
independent samples t test is a statistical method used to determine if there are significant
differences between the means of two independent groups (Warner, 2020). For the test to be
valid, several assumptions have to be met. These assumptions include Y scores are quantitative,
independence of observation, homogeneity of variance, and normality of population. The first
assumption, the Y score must be quantitative means that the Y outcome variable scores must be
quantitative since scores<calculate a mean for the Y scores in each group (Warner, 2020). The
implication of violating this assumption means incorrectly applying a quantitative statistic. The t
test is a quantitative statistic that computes the mean for Y scores which is expressed as a
numerical value and would be inappropriate to use with categorical variables (qualitative
statistics) (Warner, 2020). The second assumption, independence of observation, means Y scores
should also be independent of each other within groups. Scores may not be independent if
subjects in the same treatment condition are assessed in groups or pairs, if they are roommates or
partners, or if they have the chance to affect one another's behavior. In other words, the violation
that would be implicated is the opportunity for influencing each other when testing in groups or
pairs, making the study results not independent of each other (Warner, 2020). Take this
hypothetical study, for instance, students consumed high-carbohydrate or high-protein drinks and
rated their moods. “To collect data quickly, the student tested them in groups. One participant
threw up the protein shake, an event that probably affected the mood of others who were present.
COMPARING TWO INDEPENDENT GROUPS 3
This made Y scores dependent (related) within the testing groups. Testing participants
individually would have avoided this problem” (Warner, 2020, p. 332).
The third assumption is about the population distributions of Y is called homogeneity of
variance (Warner, 2020). The variances of the Y scores should be equal or homogeneous in the
two populations that correspond to the samples compared in the study (Warner, 2020) Even
though their average values vary, all of the groups under comparison have similar data spread
(Zhu, Zu, & Wong, 2024). By comparing the sample variances from scores within the two
groups using the Levene F test, one can determine whether this sample is violated (Warner,
2020). An illustration of this would be evaluating the impact of three distinct treatment
approaches on the treatment outcomes of clients using the F test. In To prevent Type 1 errors,
homogeneity of variance is essential (Warner, 2020). These tests may no longer be valid and, as
a result, may not yield the appropriate statistical inference when the variances of responses from
several groups are unequal (Zhu, Zu, & Wong, 2024). Homogeneity of variance violations
happen when the calculated test statistic stops being reliable. This makes it more likely that a
correct null hypothesis will be wrongly rejected, leading to a "false positive" (Type 1 error)
(Warner, 2020; Zhu, Zu, & Wong, 2024). When variances of responses from patients receiving
various treatments appear different, a common strategy is to use alternative statistical tests that
do not require variance homogeneity are derived such as the equal variances not assumed t
(Warner, 2020; Zhu, Zu, & Wong, 2024).
The fourth assumption is Y scores be evaluated from normally distributed populations
(Warner, 2020). Sample size is crucial in assessing normality of population. A large sample size
allows for a more accurate representation of the population distribution; whereas, in a small
sample size normality is not guaranteed (Kim & Park, 2019). A small sample size can also
COMPARING TWO INDEPENDENT GROUPS 4
increase the presence of outliers as fewer data points to analyze increases the probability of
extreme values affecting the research (Popukaylo, 2016). In fact, outliers violating the
assumption of normality (Warner, 2020). Earlier readings for this course stated that<the means
are insufficiently robust against outlier violations since they may affect the estimated means,
which could result in inaccurate results when evaluating test statistics (Warner, 2020).
Additionally, the existence of extreme outliers can indicate that the data is skewed from the
normal distribution, which is a crucial t test assumption (Warner, 2020). The presence of outliers
in each group can be assessed using a boxplot or histogram of Y scores, separately for each group
(Warner, 2020). Running studies like t tests once with outliers included and once with them
deleted can help a beginner learn how things differ (Warner, 2020).
Among the assumptions mentioned above to be met for the the independent-samples t test
it is important to recognize that the presence of outliers may result in p values that either
overestimate or underestimate the actual risk for Type I error (Warner, 2020). As already stated
in previous assignments p values is the likelihood that the data measured was by chance, the null
hypothesis (Warner, 2020). In contrast, a t test compares the means between two groups
(Warner, 2020). With this being said a low p value from a statistics test suggests a statistical
significance between two groups (Warner, 2020). The p value default for a p value is set at .05
(Warner, 2020). A low p value indicates a high likelihood of rejecting the null hypothesis even
when it is actually true, which is the definition of a Type 1 error. Put differently, a low p value
implies that the observed data is unlikely to have occurred even if the null hypothesis is true
which results to rejecting the null hypothesis which is rare in research. Rejecting the null
hypothesis is rare in research because the null hypothesis is designed to state that there is no
COMPARING TWO INDEPENDENT GROUPS 5
difference and strong statistical evidence needs to exist to prove that it was not a rare variation.
Therefore, though the risk can be zero it can never exactly be zero (Warner, 2020).
There are Several Indices on Effect Sizes for Independent Samples T Tests. Describe
Three of These and When One Might be Used Over the Others. Next, Give a Situation in
Which a Research Reports a Large Eta Squared Size (eta squared = .64), Why Might Their
Reported T Value Be Small and Not Statistically Significant? What May Be Inference
From Such a Situation? Indicate and Provide Examples of Three of the Factors that
Influence the Size of T.
This next section focuses on effect sizes for independent t samples. While a p value can
inform the reader whether an effect exists, the value alone will not reveal the size of the effect
(Betensky, 2019). There are a number of effect size indices; the three indices for the independent
samples t test that will be covered are point M1-M2, Cohen's D, and point biserial. When
discussing effect sizes, M1-M2 denotes the difference between two group means, where M1 is
the group 1 mean and M2 is the group mean (Warner, 2020). Sometimes, the difference in means
can be regarded as information regarding every day, clinical, or practical significance. In this
hypothetical scenario, individuals who drank 150 mg of caffeine, or roughly one cup of coffee,
had heart rates that were roughly 10 beats per minute greater than those of those who did not.
Although there is a discernible change, it is not significant enough for people to be concerned
(Warner, 2020). M1 - M2 is a core element in calculating Cohen's D. Cohen’s D is<a popular
effect size metric, it<is primarily calculated by subtracting the smaller mean from the larger mean
and dividing the result by the pooled standard deviation of both groups. This yields a
standardized measure of the effect size between the two groups (Warner, 2020). Finally, point
biserial, is another name for Pearson r correlation as it shows the relationship between a
continuous variable such as height and a dichotomous variable like six (Warner, 2020). One
index of effect size should be included with every test for statistical significance (Warner, 2020).
Cohen’s d is commonly used over other effect sizes indexes as it compares the difference
COMPARING TWO INDEPENDENT GROUPS 6
between two groups by comparing the difference in standard deviations, making it useful for
clinical, practical, and rea-world situations (Warner, 2020).
So far in this essay Y scores have been discussed. Y scores in the context of t scores
speaks to individual data points in the group to be analyzed where y represents the variable to
analyzed and a t test is used to compare the means of two groups to determine if the observed
difference between the two groups is statistically significant (Warner, 2020). Eta squared size
refers to the estimate of proportion of variance in a dependent variable Y that is explained by an
independent variable x (Warner, 2020). A large (eta squared = .64) indicates that the independent
variable has a substantial impact on the dependent variable, accounting for 64% of the variance
in the dependent variable (Warner, 2020). Researchers commonly use ETA sizes to do power
analysis for future similar studies such as a study examining the effects of solution focused brief
therapy techniques on depression levels with the solution focused technique having variable
showing to have an impact on the depression variable. A small and non-statistically significant t
value suggests that the observed relationship between the independent and dependent variables
may not be reliable or generalizable to the broader population. It can be then inferred that is little
to no statistical difference to the groups being compared in a statistical test. The apparent
discrepancy can arise from several factors that influence the size of the t statistic including
sample size. Another factor includes variability of data. Higher variability in the data, meaning
the data points are more spread out, decreases the t value. This is because variability makes it
more difficult to distinguish the true effect of the independent variable from the variability in the
data itself. Finally, effect size. While eta squared measures the proportion of variance explained
by the independent variable, the t value also considers the difference between the means of the
COMPARING TWO INDEPENDENT GROUPS 8
References
Andrade C. (2019). The<p<value and statistical significance: misunderstandings, explanations,
challenges, and alternatives. Indian Journal of Psychological Medicine, 41(3):210-215.
https://doi: 10.4103/IJPSYM.IJPSYM_193_19
Betensky, R.A. (2019). The p value requires context, not a threshold. The American Statistician,
73(1), 15-117. https://doi.org/10.108/00031305.2018.1529624
Kim, T.K., & Park, J.H. (2019) More about the basic assumptions of t-test: normality and sample
size. Korean Journal of Anesthesiology, 72(4):331-335. https://doi:
10.4097/kja.d.18.00292
Patino, C.M. & Ferreira, J.C. (2018). Meeting the assumptions of statistical tests: an important
and often forgotten step to reporting valid results. The Brazilian Journal of Pneumology,
44(5), 353. https://doi: 10.1590/S1806-37562018000000303
Popukaylo, V.S. (2016). Detection of outliers in processing of small size data. Tekhnologiya i
Konstruirovanie v Elektronnoi Apparature, 4(5). 42-46.
https://doi.org/10.15222/TKEA2016.4-5.42
Warner, R. M. (2020).<Applied Statistics I<(3rd ed.). SAGE Publications, Inc
Zhou Y, Zhu Y, Wong WK. Statistical tests for homogeneity of variance for clinical trials and
recommendations. (2023). Contemporary Clinical Trials Communication, 33, 101119.
https://doi: 10.1016/j.conctc.2023.101119
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