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How to Report T-Test Results (With Examples)

We can use the following general format to report the results of a one sample t-test :

A one sample t-test was performed to compare [variable of interest] against the population mean.   The mean value of [variable of interest] (M = [Mean], SD = [standard deviation]) was significantly [higher, lower, or different] than the population mean; t(df) = [t-value], p = [p-value].

We can use the following format to report the results of an independent two samples t-test :

A two sample t-test was performed to compare [response variable of interest] in [group 1] and [group 2].   There [was or was not] a significant difference in [response variable of interest] between [group1] (M = [Mean], SD = [standard deviation]) and [group2] (M = [Mean], SD = [standard deviation]); t(df) = [t-value], p = [p-value].

We can use the following format to report the results of a paired samples t-test :

A paired samples t-test was performed to compare [response variable of interest] in [group 1] and [group 2].   There [was or was not] a significant difference in [response variable of interest] between [group1] (M = [Mean], SD = [standard deviation]) and [group2] (M = [Mean], SD = [standard deviation]); t(df) = [t-value], p = [p-value].

Note: The “M” in the results stands for sample mean, the “SD” stands for sample standard deviation, and “df” stands for degrees of freedom associated with the t-test statistic.

The following examples show how to report the results of each type of t-test in practice.

Example: Reporting Results of a One Sample T-Test

A botanist wants to know if the mean height of a certain species of plant is equal to 15 inches. She collects a random sample of 12 plants and performs a one sample-test.

The following screenshot shows the results of the test:

Here’s how to report the results of the test:

A one sample t-test was performed to compare the mean height of a certain species of plant against the population mean.   The mean value of height (M = 14.33, SD = 1.37) was not significantly different than the population mean; t(11) = -1.685, p = .120.

Example: Reporting Results of an Independent Samples T-Test

Researchers want to know if a new fuel treatment leads to a change in the average miles per gallon of a certain car. To test this, they conduct an experiment in which 12 cars receive the new fuel treatment and 12 cars do not.

The following screenshot shows the results of the independent samples t-test:

A two sample t-test was performed to compare miles per gallon between fuel treatment and no fuel treatment.   There was not a significant difference in miles per gallon between fuel treatment (M = 22.75, SD = 3.25) and no fuel treatment (M = 21, SD = 2.73); t(22) = -1.428, p = .167.

Example: Reporting Results of a Paired Samples T-Test

Researchers want to know if a new fuel treatment leads to a change in the average mpg of a certain car. To test this, they conduct an experiment in which they measure the mpg of 12 cars with and without the fuel treatment.

The following screenshot shows the results of the paired samples t-test:

A paired samples t-test was performed to compare miles per gallon between fuel treatment and no fuel treatment.   There was a significant difference in miles per gallon between fuel treatment (M = 22.75, SD = 3.25) and no fuel treatment (M = 21, SD = 2.73); t(11) = -2.244, p = .046.

Additional Resources

Use the following calculators to automatically perform various t-tests:

One Sample t-test Calculator Two Sample t-test Calculator Paired Samples t-test Calculator

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• Knowledge Base

An Introduction to t Tests | Definitions, Formula and Examples

Published on January 31, 2020 by Rebecca Bevans . Revised on June 22, 2023.

A t test is a statistical test that is used to compare the means of two groups. It is often used in hypothesis testing to determine whether a process or treatment actually has an effect on the population of interest, or whether two groups are different from one another.

• The null hypothesis ( H 0 ) is that the true difference between these group means is zero.
• The alternate hypothesis ( H a ) is that the true difference is different from zero.

Table of contents

When to use a t test, what type of t test should i use, performing a t test, interpreting test results, presenting the results of a t test, other interesting articles, frequently asked questions about t tests.

A t test can only be used when comparing the means of two groups (a.k.a. pairwise comparison). If you want to compare more than two groups, or if you want to do multiple pairwise comparisons, use an   ANOVA test  or a post-hoc test.

The t test is a parametric test of difference, meaning that it makes the same assumptions about your data as other parametric tests. The t test assumes your data:

• are independent
• are (approximately) normally distributed
• have a similar amount of variance within each group being compared (a.k.a. homogeneity of variance)

If your data do not fit these assumptions, you can try a nonparametric alternative to the t test, such as the Wilcoxon Signed-Rank test for data with unequal variances .

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When choosing a t test, you will need to consider two things: whether the groups being compared come from a single population or two different populations, and whether you want to test the difference in a specific direction.

One-sample, two-sample, or paired t test?

• If the groups come from a single population (e.g., measuring before and after an experimental treatment), perform a paired t test . This is a within-subjects design .
• If the groups come from two different populations (e.g., two different species, or people from two separate cities), perform a two-sample t test (a.k.a. independent t test ). This is a between-subjects design .
• If there is one group being compared against a standard value (e.g., comparing the acidity of a liquid to a neutral pH of 7), perform a one-sample t test .

One-tailed or two-tailed t test?

• If you only care whether the two populations are different from one another, perform a two-tailed t test .
• If you want to know whether one population mean is greater than or less than the other, perform a one-tailed t test.
• Your observations come from two separate populations (separate species), so you perform a two-sample t test.
• You don’t care about the direction of the difference, only whether there is a difference, so you choose to use a two-tailed t test.

The t test estimates the true difference between two group means using the ratio of the difference in group means over the pooled standard error of both groups. You can calculate it manually using a formula, or use statistical analysis software.

T test formula

The formula for the two-sample t test (a.k.a. the Student’s t-test) is shown below.

In this formula, t is the t value, x 1 and x 2 are the means of the two groups being compared, s 2 is the pooled standard error of the two groups, and n 1 and n 2 are the number of observations in each of the groups.

A larger t value shows that the difference between group means is greater than the pooled standard error, indicating a more significant difference between the groups.

You can compare your calculated t value against the values in a critical value chart (e.g., Student’s t table) to determine whether your t value is greater than what would be expected by chance. If so, you can reject the null hypothesis and conclude that the two groups are in fact different.

T test function in statistical software

Most statistical software (R, SPSS, etc.) includes a t test function. This built-in function will take your raw data and calculate the t value. It will then compare it to the critical value, and calculate a p -value . This way you can quickly see whether your groups are statistically different.

In your comparison of flower petal lengths, you decide to perform your t test using R. The code looks like this:

Download the data set to practice by yourself.

Sample data set

If you perform the t test for your flower hypothesis in R, you will receive the following output:

The output provides:

• An explanation of what is being compared, called data in the output table.
• The t value : -33.719. Note that it’s negative; this is fine! In most cases, we only care about the absolute value of the difference, or the distance from 0. It doesn’t matter which direction.
• The degrees of freedom : 30.196. Degrees of freedom is related to your sample size, and shows how many ‘free’ data points are available in your test for making comparisons. The greater the degrees of freedom, the better your statistical test will work.
• The p value : 2.2e-16 (i.e. 2.2 with 15 zeros in front). This describes the probability that you would see a t value as large as this one by chance.
• A statement of the alternative hypothesis ( H a ). In this test, the H a is that the difference is not 0.
• The 95% confidence interval . This is the range of numbers within which the true difference in means will be 95% of the time. This can be changed from 95% if you want a larger or smaller interval, but 95% is very commonly used.
• The mean petal length for each group.

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When reporting your t test results, the most important values to include are the t value , the p value , and the degrees of freedom for the test. These will communicate to your audience whether the difference between the two groups is statistically significant (a.k.a. that it is unlikely to have happened by chance).

You can also include the summary statistics for the groups being compared, namely the mean and standard deviation . In R, the code for calculating the mean and the standard deviation from the data looks like this:

flower.data %>% group_by(Species) %>% summarize(mean_length = mean(Petal.Length), sd_length = sd(Petal.Length))

In our example, you would report the results like this:

If you want to know more about statistics , methodology , or research bias , make sure to check out some of our other articles with explanations and examples.

• Chi square test of independence
• Statistical power
• Descriptive statistics
• Degrees of freedom
• Pearson correlation
• Null hypothesis

Methodology

• Double-blind study
• Case-control study
• Research ethics
• Data collection
• Hypothesis testing
• Structured interviews

Research bias

• Hawthorne effect
• Unconscious bias
• Recall bias
• Halo effect
• Self-serving bias
• Information bias

A t-test is a statistical test that compares the means of two samples . It is used in hypothesis testing , with a null hypothesis that the difference in group means is zero and an alternate hypothesis that the difference in group means is different from zero.

A t-test measures the difference in group means divided by the pooled standard error of the two group means.

In this way, it calculates a number (the t-value) illustrating the magnitude of the difference between the two group means being compared, and estimates the likelihood that this difference exists purely by chance (p-value).

Your choice of t-test depends on whether you are studying one group or two groups, and whether you care about the direction of the difference in group means.

If you are studying one group, use a paired t-test to compare the group mean over time or after an intervention, or use a one-sample t-test to compare the group mean to a standard value. If you are studying two groups, use a two-sample t-test .

If you want to know only whether a difference exists, use a two-tailed test . If you want to know if one group mean is greater or less than the other, use a left-tailed or right-tailed one-tailed test .

A one-sample t-test is used to compare a single population to a standard value (for example, to determine whether the average lifespan of a specific town is different from the country average).

A paired t-test is used to compare a single population before and after some experimental intervention or at two different points in time (for example, measuring student performance on a test before and after being taught the material).

A t-test should not be used to measure differences among more than two groups, because the error structure for a t-test will underestimate the actual error when many groups are being compared.

If you want to compare the means of several groups at once, it’s best to use another statistical test such as ANOVA or a post-hoc test.

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Independent Samples T-Test

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An independent samples t-test compares the means of two groups. The data are interval for the groups. There is not an assumption of normal distribution (if the distribution of one or both groups is really unusual, the t-test will not give good results with unequal sample sizes), but there is an assumption that the two standard deviations are equal. If the sample sizes are equal or very similar in size, even that assumption is not critical.

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How to Report T-Test Results (With Examples)

We can use the following general format to report the results of a one sample t-test :

A one sample t-test was performed to compare [variable of interest] against the population mean.   The mean value of [variable of interest] (M = [Mean], SD = [standard deviation]) was significantly [higher, lower, or different] than the population mean; t(df) = [t-value], p = [p-value].

We can use the following format to report the results of an independent two samples t-test :

A two sample t-test was performed to compare [response variable of interest] in [group 1] and [group 2].   There [was or was not] a significant difference in [response variable of interest] between [group1] (M = [Mean], SD = [standard deviation]) and [group2] (M = [Mean], SD = [standard deviation]); t(df) = [t-value], p = [p-value].

We can use the following format to report the results of a paired samples t-test :

A paired samples t-test was performed to compare [response variable of interest] in [group 1] and [group 2].   There [was or was not] a significant difference in [response variable of interest] between [group1] (M = [Mean], SD = [standard deviation]) and [group2] (M = [Mean], SD = [standard deviation]); t(df) = [t-value], p = [p-value].

Note: The “M” in the results stands for sample mean, the “SD” stands for sample standard deviation, and “df” stands for degrees of freedom associated with the t-test statistic.

The following examples show how to report the results of each type of t-test in practice.

Example: Reporting Results of a One Sample T-Test

A botanist wants to know if the mean height of a certain species of plant is equal to 15 inches. She collects a random sample of 12 plants and performs a one sample-test.

The following screenshot shows the results of the test:

Here’s how to report the results of the test:

A one sample t-test was performed to compare the mean height of a certain species of plant against the population mean.   The mean value of height (M = 14.33, SD = 1.37) was not significantly different than the population mean; t(11) = -1.685, p = .120.

Example: Reporting Results of an Independent Samples T-Test

Researchers want to know if a new fuel treatment leads to a change in the average miles per gallon of a certain car. To test this, they conduct an experiment in which 12 cars receive the new fuel treatment and 12 cars do not.

The following screenshot shows the results of the independent samples t-test:

A two sample t-test was performed to compare miles per gallon between fuel treatment and no fuel treatment.   There was not a significant difference in miles per gallon between fuel treatment (M = 22.75, SD = 3.25) and no fuel treatment (M = 21, SD = 2.73); t(22) = -1.428, p = .167.

Example: Reporting Results of a Paired Samples T-Test

Researchers want to know if a new fuel treatment leads to a change in the average mpg of a certain car. To test this, they conduct an experiment in which they measure the mpg of 12 cars with and without the fuel treatment.

The following screenshot shows the results of the paired samples t-test:

A paired samples t-test was performed to compare miles per gallon between fuel treatment and no fuel treatment.   There was a significant difference in miles per gallon between fuel treatment (M = 22.75, SD = 3.25) and no fuel treatment (M = 21, SD = 2.73); t(11) = -2.244, p = .046.

Additional Resources

Use the following calculators to automatically perform various t-tests:

One Sample t-test Calculator Two Sample t-test Calculator Paired Samples t-test Calculator

How to Calculate the Sum by Group in Excel

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Example of 2-Sample t

A healthcare consultant wants to compare the patient satisfaction ratings of two hospitals. The consultant collects ratings from 20 patients for each of the hospitals.

The consultant performs a 2-sample t-test to determine whether there is a difference in the patient ratings between the hospitals.

• Open the sample data, HospitalComparison.MTW .
• Choose Stat > Basic Statistics > 2-Sample t .
• From the drop-down list, select Both samples are in one column .
• In Samples , enter Rating .
• In Sample IDs , enter Hospital .

Interpret the results

The null hypothesis states that the difference between ratings for the two hospitals is 0. Because the p-value is 0.000, which is less than the significance level of 0.05, the consultant rejects the null hypothesis and concludes that the ratings for the two hospitals differ.

Descriptive Statistics: Rating

Estimation for difference.

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Article Contents

• Misuse of t-test because data do not obey normal distribution
• Misuse of independent sample t-test because of paired samples
• Misuse of independent sample t-test because there are more than three levels in independent samples
• Misuse of independent sample t-test because of factorial design data
• Misuse of independent sample t-test because of repeated measurement design

Acknowledgements

Authors’ contributions, availability of data and materials, supplementary information, analysis of t- test misuses and spss operations in medical research papers.

• Article contents
• Figures & tables
• Supplementary Data

Guangping Liang, Wenliang Fu, Kaifa Wang, Analysis of t- test misuses and SPSS operations in medical research papers, Burns & Trauma , Volume 7, 2019, s41038–019–0170–3, https://doi.org/10.1186/s41038-019-0170-3

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In medical research papers, the selection of appropriate statistical methods serves as one of the pivotal premises to ensure the quality of papers and credibility of their results [ 1 – 3 ]. To correctly perform the statistical analysis of quantitative data, two key points should be considered. One is to identify the type of experimental design correctly, and the other is to check whether data meets the preconditions of parameter test [ 2 – 4 ]. Otherwise, it may cause different misuse in some situations and may even draw different or opposite conclusions about the same data.

As one of the most commonly used statistical methods in medical research papers, t- test can be divided into one-sample t- test and two-sample t- test [ 3 , 4 ]. Thus, it is inappropriate to compare the means among multiple groups (more than three). Concretely, one-sample t- test is used to compare one group’s average value to a single number (a known population mean, for example, the norm). The two-sample t test is a type of inferential statistic used to determine if there is a significant difference between the means of two groups. Furthermore, there are two types of two-sample t -test [ 3 , 4 ]. One is independent sample t- test (group t- test), which is performed when the samples typically consist of independent population. The other is paired (or correlated) sample t- test, which is used when each observation in one group is paired with a related observation in the other group, i.e., the samples typically consist of matched pairs of similar units, or when there are cases of repeated measures.

Note that t- test belongs to the category of parametric test. The assumptions of the parametric test, including independence, normality, and homogeneity of variance, must be met to ensure the correct use of t- test [ 3 , 4 ]. In addition, according to the theoretical deduction of t- test, it can only be applied to the quantitative data of single factor design, so it is inappropriate to perform t- test for multifactor design. For example, there are multiple independent variables/factors (such as gender and different types and dosage of drugs) and the comparisons among groups after controlling for simple effects of each independent variable.

As a journal editor and reviewer, we often encounter that some authors blindly use t- test to process quantitative data without analyzing the prerequisites of t- test or considering the type of experimental design, especially to independent sample t- test (group t- test). In order to improve the quality of statistical analysis in medical research papers, according to the problems found in the process of reviewing manuscripts, we summarized the following five most common misuses of t- test and analyzed them with examples. We hope that it can provide real help to improve our data analysis ability.

It is particularly noted that all the examples herein are artificially constructed for the purpose of illustration and do not represent actual clinical design and data. They are only for reference in the selection of statistical analysis methods.

Misuse of t- test because data do not obey normal distribution

Normal fitting tests, including the Shapiro-Wilk test for small sample size ( n  ≤ 50) or Kolmogorov-Smirnov test for large sample size ( n  > 50), usually require the analysis of the original data. However, there is a common and concise method to judge whether the data obey normal distribution, that is, to compare the mean and corresponding standard deviation (SD) of the data. If the mean is much smaller than its standard deviation, then the data may not obey the normal distribution, so t- test may also be inappropriate. In this case, it is better to perform t- test after an appropriate variable transformation (such as logarithm transforms and rank transforms) or perform nonparametric test method for original data.

Statistical results of age between experimental group and control group

Note: Data are presented as mean ± standard deviation

The data are quantitative data for two independent samples under single factor design. However, from Table  1 , we can find that the standard deviation is larger than its mean value in control group. Thus, the age in control group may not meet normal distribution. As a result, it may be inappropriate to analyze this data by the independent sample t- test directly.

[Correction]

Since the sample size of two groups is less than 50, the Shapiro-Wilk test is more suitable for normal fitting test. Selecting “Analyze➔Descriptive Statistics➔Explore…” and ticking “Normality plots with tests” in the “Plots” dialog box in SPSS. The results show that the age in experimental group accepts the normal distribution hypothesis ( W  = 0.915, p  = 0.080), but the age in control group rejects the normal distribution hypothesis ( W  = 0.635, p  < 0.001). Therefore, appropriate variable transformation should be performed if t- test must be used. In fact, the nonparametric Wilcoxon rank sum W test is a simpler and more suitable statistical method, and the Mann-Whitney U test method should be selected in this case. Selecting “Analyze➔Nonparametric Tests➔2 Independent Samples…” and ticking “Mann-Whitney U ” in “Test Type” part. After performing the test in SPSS, we have the test statistic U  = 116.500 and p  = 0.024. As a result, we can conclude that the difference of mean rank has statistical significance between the experimental group and control group, which is completely contrary to the results of independent sample t- test (Table  1 ). By the way, when a variable does not obey the normal distribution, it is better to report as median with its corresponding first and third quartiles (Q1–Q3) or median with its range, not as mean and standard deviation. In the following parts, all variables are subject to the assumption of normal distribution without special explanation.

Misuse of independent sample t- test because of paired samples

In medical research, before-after study in the same patient is often used to compare the effect of a treatment factor (such as drug and operation). This is a typical self-matching experimental design type, which does not meet the independent assumption of independent sample t- test. In this case, the paired sample t- test is more suitable if the difference value is met normally distributed. Otherwise, the nonparametric test (Wilcoxon signed rank test) of two related samples is recommended.

Evaluation of scar area of burn patients before and after treatment

Clearly, the independent assumption of independent sample t- test is not satisfied under the study protocol, and independent sample t- test is inappropriate for the data.

Using the origin data and paired samples t- test, i.e., selecting “Analyze➔Compare Means➔Paired-Samples T Test…” in SPSS, we have the test statistic t  = 10.025 and p  < 0.001. It should be noted that in this example, by comparing the p values obtained by the two methods, we find that the result of the independent sample t- test may underestimate the efficacy of the treatment scheme, though both results indicate that the treatment scheme can significantly reduce the scar area of burn patients.

Misuse of independent sample t- test because there are more than three levels in independent samples

The single factor k -level ( k ≥ 3) independent sample design is a widely used experimental design method in medical experiments. For example, to investigate the difference of a physiological index with different disease types, we measured the index of patients with k ( k  ≥ 3) disease types. In this case, we need to compare the means among k independent samples and determine whether there are any statistically significant differences between the means of three or more independent (unrelated) groups. Because direct multiple use of independent samples t- test will increase the probability of type I error, one-way analysis of variance (ANOVA) is more suitable at this time. If the one-way ANOVA returns a statistically significant result, we accept the alternative hypothesis, which is that there are at least two group means that are statistically significantly different from each other. To determine which specific groups differed from each other, we further need to perform post hoc multiple comparisons. If we want to compare each group with the control group, Dunnett’s test is recommended.

Statistical comparison of antihypertensive effects

* means that the comparison between low-dose group and placebo group under independent samples t-test ( t  = 2.517, p  = 0.019)

### means that the comparison between high-dose group and placebo group under independent samples t-test ( t  = 6.301, p  < 0.001)

These data are typical quantitative data of multigroup independent sample design, also known as the single factor design with multiple levels, and the number of levels is 3. Thus, it is not appropriate to perform the independent sample t- test directly for comparisons with control group.

According to the study design, selecting “Analyze➔Compare Means➔One-Way ANOVA…” and ticking “Dunnett” in the “Post Hoc Multiple Comparisons” dialog box in SPSS, we perform one-way ANOVA and Dunnett’s post hoc test to compare each dose group with the placebo group. The results indicate that there is a statistically significant difference between groups as determined by one-way ANOVA ( F  = 24.728, p  < 0.001). The results of multiple comparisons show that the difference between low-dose group and placebo group is not statistically significant ( p  = 0.069), which is completely contrary to the results of the independent sample t -test (Table  3 ). The difference between high-dose group and placebo group is still statistically significant ( p  < 0.001).

Misuse of independent sample t- test because of factorial design data

To understand the effect of two or more independent variables upon a single dependent variable, completely randomized factorial design is often used in medical experiments or clinical trials. A factor is a variable that is controlled and varied during the course of an experiment. In a factorial design, there are two or more factors with multiple levels that are crossed, e.g., two dose levels of drug A and two levels of drug B can be crossed to yield a total of four treatment combinations. Factorial designs offer certain advantages over conventional designs. The design can examine not only the differences among the levels of each factor, but also the interactions among the factors. For quantitative data of factorial design, direct multiple use of independent sample t- test will not only increase the probability of type I error, but also lead to wrong conclusions when there is interaction between various factors. A more appropriate method at this point is to perform ANOVA of factorial design. Taking two factors of independent samples as an example, it is also called the two-way ANOVA of independent samples.

Comparison of pain scores of patients with three disease types and two treatment schemes

* indicates the comparison between two treatment schemes by independent samples t -test, p  < 0.05 is labeled as * and p  < 0.001 is ***

# indicates the comparison between trauma/arthritis group and burn group by independent samples t -test, p  < 0.05 is labeled as # and p  < 0.01 is ##

$ indicates the comparison between arthritis group and trauma group by independent samples t-test, p  < 0.05 is labeled as $ and p  < 0.01 is $$

This study involves two factors. One is treatment factor with two levels, scheme A and scheme B, while the other is disease type factor with three levels, burns, trauma, and arthritis. Since the patients in each level combination are different, the samples are independent. Therefore, this study belongs to the 2 × 3 factorial design, and the ANOVA of factorial design should be performed for comparative analysis. Firstly, the interaction effect between the factors should be tested. If the interaction effect is not statistically significant, the main effect of each factor can be analyzed. Otherwise, the individual effect of each factor needs to be analyzed separately.

ANOVA of factorial design should be performed using the General Linear Model in SPSS (selecting “Analyze➔General Linear Model➔Univariate…”), and the results show that the interaction term between treatment and disease type reaches the significance level ( p  < 0.001), indicating that the interaction of these two factors does have an effect on the dependent variable (pain score). Therefore, it is necessary to conduct simple primary effect test for each factor. Since these two factors are independent samples, the “Split File” instruction under drop-down menu of “Data” in SPSS can be used to select qualified samples for independent sample one-way ANOVA. Through the test, in the case of scheme B, we find that there is no statistically significant difference in the pain scores between the burn/trauma and arthritis ( p  = 0.067/0.187), which is completely contrary to the results of independent sample t- test (Table  4 ).

Misuse of independent sample t- test because of repeated measurement design

Repeated measurement designs are commonly used in longitudinal studies, such as the dynamic changes over time of temperature, blood pressure, and other indicators, which is often encountered in medical research. The purpose is usually to detect whether there is a statistical significance in the difference of the indicator values at different time points. In practice, many authors usually calculate the mean and standard deviation of each time point, and then carry out independent sample t- test repeatedly for each time point. However, according to the design principle, we know that repeated measures design uses the same subjects with every condition of the research, including the control. Thus, the measurements at different time points are correlated with each other, that is, the samples at different time points are not independent of each other. Roughly speaking, such data are often time-dependent. In this case, the appropriate analysis method is ANOVA of repeated measures designs. If there is another factor with independent samples, two-way ANOVA with mixed samples is recommended.

Comparison of a certain indicator at different postoperative time points

* indicates the comparison between 1 week after operation and other time points by independent samples t -test, p  < 0.05 is labeled as *, p  < 0.01 is ** and p  < 0.001 is ***

# indicates the comparison between 2 week after operation and other time points by independent samples t -test, p  < 0.05 is labeled as # , p  < 0.01 is ## and p  < 0.001 is ###

$ indicates the comparison between 4 week after operation and other time points by independent samples t -test, p  < 0.05 is labeled as $ , p  < 0.01 is $$ and p  < 0.001 is $$$

According to the experimental process of this study, the indicators of each patient are repeatedly measured at 1 week, 2 weeks, 4 weeks, and 8 weeks after the surgery, so the postoperative time serves as a factor of repeated measurement with four levels. In addition, gender is another factor, which is an independent sample at each level. Thus, the overall design was separated by pairwise comparison at different time points through independent sample t- test and fails to take into account the fact that the data on the same subject at different time points are not independent.

Two-way ANOVA with mixed samples should be performed using the General Linear Model in SPSS (selecting “Analyze➔General Linear Model➔Repeated Measures…”). Similarly, since the interaction between gender and postoperative time reaches the level of significance ( p  < 0.001), it is necessary to perform simple primary effect test. However, since the gender factor is an independent sample and the postoperative time factor is a related sample, the test methods for the two factors are different. For gender factor, four independent sample one-way ANOVA analyses were performed based on the four levels of postoperative time, but for postoperative time factor, two related sample ANOVAs were carried out based on the two levels of gender. Using the original data, we can find that the difference between 1 week after operation and 8 weeks after operation is not statistically significant in males ( p  = 0.057), but there is a significant difference between 8 weeks after operation and 1 week after operation in females ( p  = 0.045), which was completely contrary to the results of independent sample t- test (Table  5 ).

In summary, in order to effectively reduce misuse of statistical methods and improve credibility of the statistical results, it is necessary to carefully consider the experimental design type, distribution characteristics of the data, and other relevant factors. Concretely, we should meticulously review the applicable preconditions of each statistical analysis technique and reasonably select the appropriate method before analysis of quantitative data. In this paper, the five cases of most commonly misused t- tests are summarized, with the causes of each misuse analyzed and the more appropriate statistical methods are also offered in SPSS. By doing so, we believe that this paper can be helpful to the writing and editing of biomedical research papers.

Not applicable.

The conception and design was developed by GL and KW. The article drafting and revising were performed by WF and KW. The data analysis and interpretation, revision, and final approval of article were carried out by WF, GL, and KW.

Not applicable

All artificially constructed data are presented in the tables and additional files.

Ethics approval and consent to participate

Consent for publication, competing interests.

The authors declare that they have no competing interests.

Bahar B , Pambuccian SE , Barkan GA , Akdas Y . The use and misuse of statistical methods in cytopathology studies: review of 6 journals . Laboratory Medicine . 2019 ; 50 ( 1 ): 8 – 15 . 10.1093/labmed/lmy036 .

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Hall JC , Hill D , Watts JM . Misuse of statistical methods in the Australasian surgical literature . Aust N Z J Surg . 1982 ; 52 ( 5 ): 541 – 543 . 10.1111/j.1445-2197.1982.tb06050.x .

Gore SM , Jones IG , Rytter EC . Misuse of statistical methods: critical assessment of articles in BMJ from January to March 1976 . Br Med J . 1977 ; 1 ( 6053 ): 85 – 87 . 10.1136/bmj.1.6053.85 .

Skaik Y . The bread and butter of statistical analysis “t-test”: uses and misuses . Pak J Med Sci . 2015 ; 31 ( 6 ): 1558 – 1559 4744321

Additional file 1: The original data of the Example 1.

Additional file 2: The original data of the Example 2.

Additional file 3: The original data of the Example 3.

Additional file 4: The original data of the Example 4.

Additional file 5: The original data of the Example 5.

Analysis of variance

Standard deviation

Supplementary information accompanies this paper at 10.1186/s41038-019-0170-3.

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Analysis of t- test misuses and SPSS operations in medical research papers

Guangping liang.

1 State Key Laboratory of Trauma, Burns, and Combined Injury, Institute of Burn Research, Southwest Hospital, Third Military Medical University (Army Medical University), Chongqing, 400038 People’s Republic of China

Wenliang Fu

2 School of Mathematics and Statistics, Southwest University, Chongqing, 400715 People’s Republic of China

Associated Data

All artificially constructed data are presented in the tables and additional files.

In medical research papers, the selection of appropriate statistical methods serves as one of the pivotal premises to ensure the quality of papers and credibility of their results [ 1 – 3 ]. To correctly perform the statistical analysis of quantitative data, two key points should be considered. One is to identify the type of experimental design correctly, and the other is to check whether data meets the preconditions of parameter test [ 2 – 4 ]. Otherwise, it may cause different misuse in some situations and may even draw different or opposite conclusions about the same data.

As one of the most commonly used statistical methods in medical research papers, t- test can be divided into one-sample t- test and two-sample t- test [ 3 , 4 ]. Thus, it is inappropriate to compare the means among multiple groups (more than three). Concretely, one-sample t- test is used to compare one group’s average value to a single number (a known population mean, for example, the norm). The two-sample t test is a type of inferential statistic used to determine if there is a significant difference between the means of two groups. Furthermore, there are two types of two-sample t -test [ 3 , 4 ]. One is independent sample t- test (group t- test), which is performed when the samples typically consist of independent population. The other is paired (or correlated) sample t- test, which is used when each observation in one group is paired with a related observation in the other group, i.e., the samples typically consist of matched pairs of similar units, or when there are cases of repeated measures.

Note that t- test belongs to the category of parametric test. The assumptions of the parametric test, including independence, normality, and homogeneity of variance, must be met to ensure the correct use of t- test [ 3 , 4 ]. In addition, according to the theoretical deduction of t- test, it can only be applied to the quantitative data of single factor design, so it is inappropriate to perform t- test for multifactor design. For example, there are multiple independent variables/factors (such as gender and different types and dosage of drugs) and the comparisons among groups after controlling for simple effects of each independent variable.

As a journal editor and reviewer, we often encounter that some authors blindly use t- test to process quantitative data without analyzing the prerequisites of t- test or considering the type of experimental design, especially to independent sample t- test (group t- test). In order to improve the quality of statistical analysis in medical research papers, according to the problems found in the process of reviewing manuscripts, we summarized the following five most common misuses of t- test and analyzed them with examples. We hope that it can provide real help to improve our data analysis ability.

It is particularly noted that all the examples herein are artificially constructed for the purpose of illustration and do not represent actual clinical design and data. They are only for reference in the selection of statistical analysis methods.

Misuse of t- test because data do not obey normal distribution

Normal fitting tests, including the Shapiro-Wilk test for small sample size ( n  ≤ 50) or Kolmogorov-Smirnov test for large sample size ( n  > 50), usually require the analysis of the original data. However, there is a common and concise method to judge whether the data obey normal distribution, that is, to compare the mean and corresponding standard deviation (SD) of the data. If the mean is much smaller than its standard deviation, then the data may not obey the normal distribution, so t- test may also be inappropriate. In this case, it is better to perform t- test after an appropriate variable transformation (such as logarithm transforms and rank transforms) or perform nonparametric test method for original data.

A researcher adopts the independent sample t- test to compare the demographic data (age) between the experimental group and the control group. Table  1 provides the statistical results (see Additional file 1 for the original data). Is this appropriate?

Statistical results of age between experimental group and control group

Note: Data are presented as mean ± standard deviation

The data are quantitative data for two independent samples under single factor design. However, from Table  1 , we can find that the standard deviation is larger than its mean value in control group. Thus, the age in control group may not meet normal distribution. As a result, it may be inappropriate to analyze this data by the independent sample t- test directly.

[Correction]

Since the sample size of two groups is less than 50, the Shapiro-Wilk test is more suitable for normal fitting test. Selecting “Analyze➔Descriptive Statistics➔Explore…” and ticking “Normality plots with tests” in the “Plots” dialog box in SPSS. The results show that the age in experimental group accepts the normal distribution hypothesis ( W  = 0.915, p  = 0.080), but the age in control group rejects the normal distribution hypothesis ( W  = 0.635, p  < 0.001). Therefore, appropriate variable transformation should be performed if t- test must be used. In fact, the nonparametric Wilcoxon rank sum W test is a simpler and more suitable statistical method, and the Mann-Whitney U test method should be selected in this case. Selecting “Analyze➔Nonparametric Tests➔2 Independent Samples…” and ticking “Mann-Whitney U ” in “Test Type” part. After performing the test in SPSS, we have the test statistic U  = 116.500 and p  = 0.024. As a result, we can conclude that the difference of mean rank has statistical significance between the experimental group and control group, which is completely contrary to the results of independent sample t- test (Table  1 ). By the way, when a variable does not obey the normal distribution, it is better to report as median with its corresponding first and third quartiles (Q1–Q3) or median with its range, not as mean and standard deviation. In the following parts, all variables are subject to the assumption of normal distribution without special explanation.

Misuse of independent sample t- test because of paired samples

In medical research, before-after study in the same patient is often used to compare the effect of a treatment factor (such as drug and operation). This is a typical self-matching experimental design type, which does not meet the independent assumption of independent sample t- test. In this case, the paired sample t- test is more suitable if the difference value is met normally distributed. Otherwise, the nonparametric test (Wilcoxon signed rank test) of two related samples is recommended.

In order to explore the effect of a certain treatment scheme on the scar of burn patients, the scar area of the patients is measured 1 day before and 1 week after treatment, respectively. And the independent sample t- test is used to compare the changes of scar area of the patients before and after treatment. Table  2 shows the statistical results (see Additional file 2 for the original data). Is this appropriate?

Evaluation of scar area of burn patients before and after treatment

Clearly, the independent assumption of independent sample t- test is not satisfied under the study protocol, and independent sample t- test is inappropriate for the data.

Using the origin data and paired samples t- test, i.e., selecting “Analyze➔Compare Means➔Paired-Samples T Test…” in SPSS, we have the test statistic t  = 10.025 and p  < 0.001. It should be noted that in this example, by comparing the p values obtained by the two methods, we find that the result of the independent sample t- test may underestimate the efficacy of the treatment scheme, though both results indicate that the treatment scheme can significantly reduce the scar area of burn patients.

Misuse of independent sample t- test because there are more than three levels in independent samples

The single factor k -level ( k ≥ 3) independent sample design is a widely used experimental design method in medical experiments. For example, to investigate the difference of a physiological index with different disease types, we measured the index of patients with k ( k  ≥ 3) disease types. In this case, we need to compare the means among k independent samples and determine whether there are any statistically significant differences between the means of three or more independent (unrelated) groups. Because direct multiple use of independent samples t- test will increase the probability of type I error, one-way analysis of variance (ANOVA) is more suitable at this time. If the one-way ANOVA returns a statistically significant result, we accept the alternative hypothesis, which is that there are at least two group means that are statistically significantly different from each other. To determine which specific groups differed from each other, we further need to perform post hoc multiple comparisons. If we want to compare each group with the control group, Dunnett’s test is recommended.

For a new antihypertensive drug, we hope to compare the antihypertensive effect of high- and low-dose groups with that of placebo group. The independent sample t- test is adopted to compare the low-dose group with the placebo group and the high-dose group with the placebo group, respectively. The statistic results are presented in Table  3 (see Additional file 3 for the original data). Is this appropriate?

Statistical comparison of antihypertensive effects

* means that the comparison between low-dose group and placebo group under independent samples t-test ( t  = 2.517, p  = 0.019)

### means that the comparison between high-dose group and placebo group under independent samples t-test ( t  = 6.301, p  < 0.001)

These data are typical quantitative data of multigroup independent sample design, also known as the single factor design with multiple levels, and the number of levels is 3. Thus, it is not appropriate to perform the independent sample t- test directly for comparisons with control group.

According to the study design, selecting “Analyze➔Compare Means➔One-Way ANOVA…” and ticking “Dunnett” in the “Post Hoc Multiple Comparisons” dialog box in SPSS, we perform one-way ANOVA and Dunnett’s post hoc test to compare each dose group with the placebo group. The results indicate that there is a statistically significant difference between groups as determined by one-way ANOVA ( F  = 24.728, p  < 0.001). The results of multiple comparisons show that the difference between low-dose group and placebo group is not statistically significant ( p  = 0.069), which is completely contrary to the results of the independent sample t -test (Table  3 ). The difference between high-dose group and placebo group is still statistically significant ( p  < 0.001).

Misuse of independent sample t- test because of factorial design data

To understand the effect of two or more independent variables upon a single dependent variable, completely randomized factorial design is often used in medical experiments or clinical trials. A factor is a variable that is controlled and varied during the course of an experiment. In a factorial design, there are two or more factors with multiple levels that are crossed, e.g., two dose levels of drug A and two levels of drug B can be crossed to yield a total of four treatment combinations. Factorial designs offer certain advantages over conventional designs. The design can examine not only the differences among the levels of each factor, but also the interactions among the factors. For quantitative data of factorial design, direct multiple use of independent sample t- test will not only increase the probability of type I error, but also lead to wrong conclusions when there is interaction between various factors. A more appropriate method at this point is to perform ANOVA of factorial design. Taking two factors of independent samples as an example, it is also called the two-way ANOVA of independent samples.

To study the difference of pain score between patients with different disease types (burn, trauma, and arthritis) after receiving two treatment schemes (named as scheme A and scheme B), ten patients were recruited for each type of disease and randomly assigned to the possible treatment schemes with equal possibility. For the measured pain scores, independent sample t- tests are performed repeatedly to compare the difference between disease types and treatment schemes. Table  4 shows the statistical results (see Additional file 4 for the original data). Is this appropriate?

Comparison of pain scores of patients with three disease types and two treatment schemes

* indicates the comparison between two treatment schemes by independent samples t -test, p  < 0.05 is labeled as * and p  < 0.001 is ***

# indicates the comparison between trauma/arthritis group and burn group by independent samples t -test, p  < 0.05 is labeled as # and p  < 0.01 is ##

$ indicates the comparison between arthritis group and trauma group by independent samples t-test, p  < 0.05 is labeled as $ and p  < 0.01 is $$

This study involves two factors. One is treatment factor with two levels, scheme A and scheme B, while the other is disease type factor with three levels, burns, trauma, and arthritis. Since the patients in each level combination are different, the samples are independent. Therefore, this study belongs to the 2 × 3 factorial design, and the ANOVA of factorial design should be performed for comparative analysis. Firstly, the interaction effect between the factors should be tested. If the interaction effect is not statistically significant, the main effect of each factor can be analyzed. Otherwise, the individual effect of each factor needs to be analyzed separately.

ANOVA of factorial design should be performed using the General Linear Model in SPSS (selecting “Analyze➔General Linear Model➔Univariate…”), and the results show that the interaction term between treatment and disease type reaches the significance level ( p  < 0.001), indicating that the interaction of these two factors does have an effect on the dependent variable (pain score). Therefore, it is necessary to conduct simple primary effect test for each factor. Since these two factors are independent samples, the “Split File” instruction under drop-down menu of “Data” in SPSS can be used to select qualified samples for independent sample one-way ANOVA. Through the test, in the case of scheme B, we find that there is no statistically significant difference in the pain scores between the burn/trauma and arthritis ( p  = 0.067/0.187), which is completely contrary to the results of independent sample t- test (Table  4 ).

Misuse of independent sample t- test because of repeated measurement design

Repeated measurement designs are commonly used in longitudinal studies, such as the dynamic changes over time of temperature, blood pressure, and other indicators, which is often encountered in medical research. The purpose is usually to detect whether there is a statistical significance in the difference of the indicator values at different time points. In practice, many authors usually calculate the mean and standard deviation of each time point, and then carry out independent sample t- test repeatedly for each time point. However, according to the design principle, we know that repeated measures design uses the same subjects with every condition of the research, including the control. Thus, the measurements at different time points are correlated with each other, that is, the samples at different time points are not independent of each other. Roughly speaking, such data are often time-dependent. In this case, the appropriate analysis method is ANOVA of repeated measures designs. If there is another factor with independent samples, two-way ANOVA with mixed samples is recommended.

To study the difference for a certain indicator at different postoperative time points, 10 patients (5 males and 5 females) are enrolled in the study and the indictor of each of them is measured at 1, 2, 4, and 8 weeks after the operation. The researchers use the independent sample t- test to analyze the difference of this indictor of different time points. The statistical results are presented in Table  5 (see Additional file 5 for the original data). Is this appropriate?

Comparison of a certain indicator at different postoperative time points

* indicates the comparison between 1 week after operation and other time points by independent samples t -test, p  < 0.05 is labeled as *, p  < 0.01 is ** and p  < 0.001 is ***

# indicates the comparison between 2 week after operation and other time points by independent samples t -test, p  < 0.05 is labeled as # , p  < 0.01 is ## and p  < 0.001 is ###

$ indicates the comparison between 4 week after operation and other time points by independent samples t -test, p  < 0.05 is labeled as $ , p  < 0.01 is $$ and p  < 0.001 is $$$

According to the experimental process of this study, the indicators of each patient are repeatedly measured at 1 week, 2 weeks, 4 weeks, and 8 weeks after the surgery, so the postoperative time serves as a factor of repeated measurement with four levels. In addition, gender is another factor, which is an independent sample at each level. Thus, the overall design was separated by pairwise comparison at different time points through independent sample t- test and fails to take into account the fact that the data on the same subject at different time points are not independent.

Two-way ANOVA with mixed samples should be performed using the General Linear Model in SPSS (selecting “Analyze➔General Linear Model➔Repeated Measures…”). Similarly, since the interaction between gender and postoperative time reaches the level of significance ( p  < 0.001), it is necessary to perform simple primary effect test. However, since the gender factor is an independent sample and the postoperative time factor is a related sample, the test methods for the two factors are different. For gender factor, four independent sample one-way ANOVA analyses were performed based on the four levels of postoperative time, but for postoperative time factor, two related sample ANOVAs were carried out based on the two levels of gender. Using the original data, we can find that the difference between 1 week after operation and 8 weeks after operation is not statistically significant in males ( p  = 0.057), but there is a significant difference between 8 weeks after operation and 1 week after operation in females ( p  = 0.045), which was completely contrary to the results of independent sample t- test (Table  5 ).

In summary, in order to effectively reduce misuse of statistical methods and improve credibility of the statistical results, it is necessary to carefully consider the experimental design type, distribution characteristics of the data, and other relevant factors. Concretely, we should meticulously review the applicable preconditions of each statistical analysis technique and reasonably select the appropriate method before analysis of quantitative data. In this paper, the five cases of most commonly misused t- tests are summarized, with the causes of each misuse analyzed and the more appropriate statistical methods are also offered in SPSS. By doing so, we believe that this paper can be helpful to the writing and editing of biomedical research papers.

Supplementary information

Acknowledgements.

Not applicable.

Abbreviations

Authors’ contributions.

The conception and design was developed by GL and KW. The article drafting and revising were performed by WF and KW. The data analysis and interpretation, revision, and final approval of article were carried out by WF, GL, and KW.

Not applicable

Availability of data and materials

Ethics approval and consent to participate, consent for publication, competing interests.

The authors declare that they have no competing interests.

Contributor Information

Guangping Liang, Email: moc.361@gggnailgnipgnaug .

Wenliang Fu, Email: moc.qq@054126568 .

Kaifa Wang, Phone: +86 15523803972, Email: moc.361@27gnawfk .

Supplementary information accompanies this paper at 10.1186/s41038-019-0170-3.