The unpaired t-test<\/strong> is used to compare the mean of two independent groups. It\u2019s also known as:<\/p>\n For example, you might want to compare the average weights of individuals grouped by gender: male and female groups, which are two unrelated\/independent groups.<\/p>\n As a rule of thumb, your study should have six or more participants in each group in order to proceed with an unpaired t-test, but ideally you would have more. An independent-samples t-test will run with less than six participants, but your ability to generalize to a larger population will be more difficult.<\/p>\n<\/div>\n The independent samples t-test comes in two different forms:<\/p>\n By default, R computes the Welch t-test, which is the safer one. The two methods give very similar results unless both the group sizes and the standard deviations are very different.<\/p>\n<\/div>\n In this article, you will learn how to:<\/p>\n Contents:<\/p>\n Make sure you have installed the following R packages:<\/p>\n Start by loading the following required packages:<\/p>\n Typical research questions are:<\/p>\n In statistics, we can define the corresponding null hypothesis (\\(H_0\\)<\/span>) as follow:<\/p>\n The corresponding alternative hypotheses (\\(H_a\\)<\/span>) are as follow:<\/p>\n Note that:<\/p>\n The independent samples t-test comes in two different forms, Student\u2019s t-test and Welch\u2019s t-test.<\/p>\n The classical Student\u2019s t-test is more restrictive. It assumes that the two groups have the same population variance.<\/p>\n \\[ where,<\/p>\n \\[ with degrees of freedom (df): \\(df = n_A + n_B - 2\\)<\/span>.<\/p>\n \\[ where, \\(S_A\\)<\/span> and \\(S_B\\)<\/span> are the standard deviation of the the two groups A and B, respectively.<\/p>\n Unlike the classic Student\u2019s t-test, the Welch t-test formula involves the variance of each of the two groups (\\(S_A^2\\)<\/span> and \\(S_B^2\\)<\/span>) being compared. In other words, it does not use the pooled variance \\(S\\)<\/span>.<\/p>\n The degrees of freedom<\/strong> of Welch t-test<\/strong> is estimated as follow :<\/p>\n \\[ A p-value can be computed for the corresponding absolute value of t-statistic (|t|).<\/p>\n If the p-value is inferior or equal to the significance level 0.05, we can reject the null hypothesis and accept the alternative hypothesis. In other words, we can conclude that the mean values of group A and B are significantly different.<\/p>\n<\/div>\n Note that, the Welch t-test is considered as the safer one. Usually, the results of the classical student\u2019s t-test<\/strong> and the Welch t-test<\/strong> are very similar unless both the group sizes and the standard deviations are very different.<\/p>\n<\/div>\n<\/div>\n Demo dataset: Load the data and show some random rows by groups:<\/p>\n Compute some summary statistics by groups: mean and sd (standard deviation)<\/p>\n Visualize the data using box plots. Plot weight by groups.<\/p>\n The two-samples independent t-test assume the following characteristics about the data:<\/p>\n In this section, we\u2019ll perform some preliminary tests to check whether these assumptions are met.<\/p>\n Outliers can be easily identified using boxplot methods, implemented in the R function There were no extreme outliers.<\/p>\n<\/div>\n Note that, in the situation where you have extreme outliers, this can be due to: 1) data entry errors, measurement errors or unusual values.<\/p>\n Yo can include the outlier in the analysis anyway if you do not believe the result will be substantially affected. This can be evaluated by comparing the result of the t-test with and without the outlier.<\/p>\n It\u2019s also possible to keep the outliers in the data and perform Wilcoxon test or robust t-test using the WRS2 package.<\/p>\n<\/div>\n<\/div>\n The normality assumption can be checked by computing the Shapiro-Wilk test for each group. If the data is normally distributed, the p-value should be greater than 0.05.<\/p>\n From the output, the two p-values are greater than the significance level 0.05 indicating that the distribution of the data are not significantly different from the normal distribution. In other words, we can assume the normality.<\/p>\n<\/div>\n You can also create QQ plots for each group. QQ plot draws the correlation between a given data and the normal distribution.<\/p>\n All the points fall approximately along the (45-degree) reference line, for each group. So we can assume normality of the data.<\/p>\n<\/div>\n Note that, if your sample size is greater than 50, the normal QQ plot is preferred because at larger sample sizes the Shapiro-Wilk test becomes very sensitive even to a minor deviation from normality.<\/p>\n Note that, in the situation where the data are not normally distributed, it\u2019s recommended to use the non parametric two-samples Wilcoxon test.<\/p>\n<\/div>\n<\/div>\n This can be done using the Levene\u2019s test. If the variances of groups are equal, the p-value should be greater than 0.05.<\/p>\n The p-value of the Levene\u2019s test is significant, suggesting that there is a significant difference between the variances of the two groups. Therefore, we\u2019ll use the Welch t-test, which doesn\u2019t assume the equality of the two variances.<\/p>\n<\/div>\n<\/div>\n<\/div>\n We want to know, whether the average weights are different between groups.<\/p>\n We\u2019ll use the pipe-friendly Recall that, by default, R computes the Welch t-test, which is the safer one. This is the test where you do not assume that the variance is the same in the two groups, which results in the fractional degrees of freedom.<\/p>\n If you want to assume the equality of variances (Student t-test), specify the option The results above show the following components:<\/p>\n Note that, you can obtain a detailed result by specifying the option Note that, to compute one tailed two samples t-test, you can specify the option \n
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t_test()<\/code> [rstatix package] will be used.<\/li>\nd<\/code> statistic redefines the difference in means as the number of standard deviations that separates those means. T-test conventional effect sizes, proposed by Cohen, are: 0.2 (small effect), 0.5 (moderate effect) and 0.8 (large effect) (Cohen 1998)<\/span>.<\/li>\n<\/ul>\n\n
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Related Book <\/a><\/h4>Practical Statistics in R II - Comparing Groups: Numerical Variables<\/div>\n
Prerequisites<\/h2>\n
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tidyverse<\/code> for data manipulation and visualization<\/li>\nggpubr<\/code> for creating easily publication ready plots<\/li>\nrstatix<\/code> provides pipe-friendly R functions for easy statistical analyses.<\/li>\ndatarium<\/code>: contains required data sets for this chapter.<\/li>\n<\/ul>\nlibrary(tidyverse)\r\nlibrary(ggpubr)\r\nlibrary(rstatix)<\/code><\/pre>\n<\/div>\nResearch questions<\/h2>\n
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Statistical hypotheses<\/h2>\n
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Formula<\/h2>\n
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\nt = \\frac{m_A - m_B}{\\sqrt{ \\frac{S^2}{n_A} + \\frac{S^2}{n_B} }}
\n\\]<\/span><\/p>\n\n
\nS^2 = \\frac{\\sum{(x-m_A)^2}+\\sum{(x-m_B)^2}}{n_A+n_B-2}
\n\\]<\/span><\/p>\n\n
\nt = \\frac{m_A - m_B}{\\sqrt{ \\frac{S_A^2}{n_A} + \\frac{S_B^2}{n_B} }}
\n\\]<\/span><\/p>\n
\ndf = (\\frac{S_A^2}{n_A}+ \\frac{S_B^2}{n_B})^2 \/ (\\frac{S_A^4}{n_A^2(n_A-1)} + \\frac{S_B^4}{n_B^2(n_B-1)} )
\n\\]<\/span><\/p>\nDemo data<\/h2>\n
genderweight<\/code> [in datarium package] containing the weight of 40 individuals (20 women and 20 men).<\/p>\n# Load the data\r\ndata(\"genderweight\", package = \"datarium\")\r\n# Show a sample of the data by group\r\nset.seed(123)\r\ngenderweight %>% sample_n_by(group, size = 2)<\/code><\/pre>\n## # A tibble: 4 x 3\r\n## id group weight\r\n## <fct> <fct> <dbl>\r\n## 1 6 F 65.0\r\n## 2 15 F 65.9\r\n## 3 29 M 88.9\r\n## 4 37 M 77.0<\/code><\/pre>\n<\/div>\nSummary statistics<\/h2>\n
genderweight %>%\r\n group_by(group) %>%\r\n get_summary_stats(weight, type = \"mean_sd\")<\/code><\/pre>\n## # A tibble: 2 x 5\r\n## group variable n mean sd\r\n## <fct> <chr> <dbl> <dbl> <dbl>\r\n## 1 F weight 20 63.5 2.03\r\n## 2 M weight 20 85.8 4.35<\/code><\/pre>\n<\/div>\nVisualization<\/h2>\n
bxp <- ggboxplot(\r\n genderweight, x = \"group\", y = \"weight\", \r\n ylab = \"Weight\", xlab = \"Groups\", add = \"jitter\"\r\n )\r\nbxp<\/code><\/pre>\n![](\"https:\/\/www.datanovia.com\/en\/wp-content\/uploads\/dn-tutorials\/r-statistics-2-comparing-groups-means\/figures\/071-unpaired-t-test-box-plot-1.png\")
<\/p>\n<\/div>\nAssumptions and preleminary tests<\/h2>\n
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Identify outliers<\/h3>\n
identify_outliers()<\/code> [rstatix package].<\/p>\ngenderweight %>%\r\n group_by(group) %>%\r\n identify_outliers(weight)<\/code><\/pre>\n## # A tibble: 2 x 5\r\n## group id weight is.outlier is.extreme\r\n## <fct> <fct> <dbl> <lgl> <lgl> \r\n## 1 F 20 68.8 TRUE FALSE \r\n## 2 M 31 95.1 TRUE FALSE<\/code><\/pre>\nCheck normality by groups<\/h3>\n
genderweight %>%\r\n group_by(group) %>%\r\n shapiro_test(weight)<\/code><\/pre>\n## # A tibble: 2 x 4\r\n## group variable statistic p\r\n## <fct> <chr> <dbl> <dbl>\r\n## 1 F weight 0.938 0.224\r\n## 2 M weight 0.986 0.989<\/code><\/pre>\nggqqplot(genderweight, x = \"weight\", facet.by = \"group\")<\/code><\/pre>\n![](\"https:\/\/www.datanovia.com\/en\/wp-content\/uploads\/dn-tutorials\/r-statistics-2-comparing-groups-means\/figures\/071-unpaired-t-test-qqplot-1.png\")
<\/p>\nCheck the equality of variances<\/h3>\n
genderweight %>% levene_test(weight ~ group)<\/code><\/pre>\n## # A tibble: 1 x 4\r\n## df1 df2 statistic p\r\n## <int> <int> <dbl> <dbl>\r\n## 1 1 38 6.12 0.0180<\/code><\/pre>\nComputation<\/h2>\n
t_test()<\/code> function [rstatix package], a wrapper around the R base function t.test()<\/code>.<\/p>\nstat.test <- genderweight %>% \r\n t_test(weight ~ group) %>%\r\n add_significance()\r\nstat.test<\/code><\/pre>\n## # A tibble: 1 x 9\r\n## .y. group1 group2 n1 n2 statistic df p p.signif\r\n## <chr> <chr> <chr> <int> <int> <dbl> <dbl> <dbl> <chr> \r\n## 1 weight F M 20 20 -20.8 26.9 4.30e-18 ****<\/code><\/pre>\nvar.equal = TRUE<\/code>:<\/p>\nstat.test2 <- genderweight %>%\r\n t_test(weight ~ group, var.equal = TRUE) %>%\r\n add_significance()\r\nstat.test2<\/code><\/pre>\n\n
.y.<\/code>: the y variable used in the test.<\/li>\ngroup1,group2<\/code>: the compared groups in the pairwise tests.<\/li>\nstatistic<\/code>: Test statistic used to compute the p-value.<\/li>\ndf<\/code>: degrees of freedom.<\/li>\np<\/code>: p-value.<\/li>\n<\/ul>\ndetailed = TRUE<\/code>.<\/p>\n<\/div>\nalternative<\/code> as follow.<\/p>\n