Here we introduce various tests & intervals in R. The following are examples of some tests & intervals in R, with more details in the pages on them.
This section has tests and intervals for one sample or one variable data. See the page on each test for more details.
| Test | Formula | Example |
|---|---|---|
| One Sample T-tests |
\(t = \frac{\bar x - \mu_0}{s/ \sqrt
n}\) and Student’s t-distribution. |
Is the mean equal to \(\mu_0\)?data = c(1.6, -2.1, 2.2, -2.4, 1.1, 3.3)t.test(data, alternative = "two.sided",mu = 0, conf.level = 0.95)
|
|
One Proportion Tests (with Z & Chi-squared) |
\(z = \frac{\frac{x}{n} - p_0}{\sqrt
\frac{p_0(1-p_0)}{n}}\) \(\text{or} \quad \chi^2_1=\sum_{i=1}^{2}\frac{(O_i-E_i)^2}{E_i}\), and standard normal distribution or chi-squared distribution. |
Is the proportion equal to \(p_0\)?prop.test(x = 55, n = 100, p = 0.5,alternative = "two.sided",conf.level = 0.95,correct = FALSE)
|
|
One
Variance Chi-squared |
\(\chi^2 =
\frac{(n-1)s^2}{\sigma_0^2}\) and chi-squared distribution. |
Is the variance equal to \(\sigma_0\)?install.packages("DescTools")library(DescTools)data = c(10.8, 8.1, 5.8, 8.8, 6.1)VarTest(data, sigma.squared = 4,alternative = "two.sided",conf.level = 0.95)
|
| One-way ANOVA |
\(F = \frac{\tt{variance \; between \;
groups}}{\tt{variance \; within \; groups}}\) \(= \frac{\left[ \sum_{j=1}^G N_j(\bar y_{j\cdot}-\bar y_{\cdot \cdot})^2 \right] / (G-1)}{\left[ \sum_{j=1}^G \sum_{k=1}^{N_j} (y_{jk}-\bar y_{j\cdot})^2 \right]/ (N-G)}\) and F-distribution. |
Are the group means equal?y = c(7.3, 5.5, 5.1, 1.3,8.5, 4.9, 6.2,5.7, 6.3, 8.2, 4.9, 6.4)groups = c("A", "A", "A", "A","B", "B", "B","C", "C", "C", "C", "C")anova(lm(y ~ groups))
|
|
Chi-squared Goodness of Fit |
\(\chi^2=\sum_{i}\frac{(O_{i}-E_{i})^2}{E_{i}}\) and chi-squared distribution. |
Does the observed frequency distribution fit the expectation? chisq.test(c(19, 16, 11, 6),p = c(0.4, 0.3, 0.2, 0.1))
|
| Exact Binomial |
The number of successes, \(x\), and binomial distribution. |
Is the proportion equal to \(p_0\)?binom.test(x = 14, n = 25, p = 0.5,alternative = "two.sided",conf.level = 0.95)
|
This section has tests and intervals for two samples or two variables data. See the page on each test for more details.
| Test | Formula | Example |
|---|---|---|
|
Welch’s Two
Sample T-tests |
\(t = \frac{(\bar x - \bar y) -
\mu_0}{\sqrt{\frac{s_x^2}{n_x} + \frac{s_y^2}{n_y}}}\) and Student’s t-distribution. |
Are the means equal, or difference equal to \(\mu_0\)?data_x = c(5.2, 2.1, 4.0, 6.3, 7.2, 8.2)data_y = c(1.4, 6.2, 3.4, 5.9,4.9, 7.1, 5.5, 4.4)t.test(data_x, data_y,alternative = "two.sided",mu = 0, conf.level = 0.95)
|
|
Pooled Two Sample T-tests (Equal Variance) |
\(t = \frac{(\bar x - \bar y) - \mu_0}{s_p
\cdot \sqrt{\frac{1}{n_x}+\frac{1}{n_y}}}\), where \(s_p = \sqrt{\frac{\left(n_x-1\right)s_{x}^2+\left(n_y-1\right)s_{y}^2}{n_x+n_y-2}}\) and Student’s t-distribution. |
Are the means equal, or difference equal to \(\mu_0\)?data_x = c(14.0, 13.1, 16.8, 11.1,9.6, 12.4, 13.6, 8.8)data_y = c(16.3, 14.8, 6.9, 17.3, 7.3)t.test(data_x, data_y, var.equal = TRUE,alternative = "less",mu = 0, conf.level = 0.95)
|
|
Paired Two Sample
T-tests (Matched Pairs) |
\(t = \frac{\bar d - d_0}{s_d/ \sqrt{n}} =
\frac{(\bar x - \bar y) - d_0}{s_d/ \sqrt{n}}\) and Student’s t-distribution. |
Is the mean of paired x and y differences equal \(d_0\)? data_x = c(8.0, 9.2, 9.1, 7.2, 9.8)data_y = c(7.5, 5.1, 8.6, 12.7, 8.4)t.test(data_x, data_y, paired = TRUE,alternative = "two.sided",mu = 0, conf.level = 0.95)
|
|
Two Proportions Tests (with Z & Chi-squared) |
\(z =
\frac{(\frac{x_1}{n_1}-\frac{x_2}{n_2})}{\sqrt {\hat p(1-\hat p)
\left(\frac{1}{n_1} + \frac{1}{n_2} \right)}}\), where \(\hat p = \frac{x_1 + x_2}{n_1 + n_2}\) \(\text{or} \quad \chi^2_1=\sum_{i,j=1}^{2}\frac{(O_{ij}-E_{ij})^2}{E_{ij}}\) and standard normal distribution or chi-squared distribution. |
Are the proportions equal?prop.test(c(x1 = 22, x2 = 33),c(n1 = 50, n2 = 60),alternative = "two.sided",conf.level = 0.95, correct = FALSE)
|
|
Pearson’s
Correlation Coefficient |
\(r = \frac{\sum ^n _{i=1}(x_i - \bar{x})(y_i
- \bar{y})}{\sqrt{\sum ^n _{i=1}(x_i - \bar{x})^2} \sqrt{\sum ^n
_{i=1}(y_i - \bar{y})^2}}\), \(t = \frac{r}{\sqrt{\frac{1 - r^2}{n-2}}}\) and Student’s t-distribution. |
Is there a linear correlation between x and y? data_x = c(8.2, 11.9, 11.1, 10.6, 9.9)data_y = c(9.5, 10.2, 8.1, 13.4, 8.8)cor.test(data_x, data_y, method = "pearson",alternative = "two.sided",conf.level = 0.95)
|
| Two Variances F-tests |
\(F =
\frac{s^2_1/\sigma^2_1}{s^2_2/\sigma^2_2}\), if \(\sigma^2_1 = \sigma^2_2\), \(F = \frac{s^2_1}{s^2_2}\) and F-distribution. |
Are the variances equal, or ratio equal to \(r_0\)?data1 = c(13.0, 9.8, 10.4, 10.5, 10.3)data2 = c(9.9, 12.3, 12.5, 7.7,11.1, 9.0, 8.1, 12.3)var.test(data1, data2, ratio = 1,alternative = "two.sided",conf.level = 0.95)
|
| Two-way ANOVA |
\(F = \frac{\tt{mean \;square \;error \;of
\;factor}}{\tt{mean \;square \;error \;of \;residuals}}\) and F-distribution. |
Are there effects from the factors?y = c(5.2, 3.1, 5.0, 7.4, 2.8, 7.2,2.0, 4.7, 4.8, 6.9, 7.0, 5.1,8.5, 8.2, 6.9, 5.3, 5.1, 5.3)factorA = c(rep("A1", 6),rep("A2", 6),rep("A3", 6))factorB = c(rep("B1", 3), rep("B2", 3),rep("B1", 3), rep("B2", 3),rep("B1", 3), rep("B2", 3))anova(lm(y ~ factorA + factorB))# For interaction effectanova(lm(y ~ factorA + factorB + factorA:factorB))
|
|
Chi-squared Contingency Table |
\(\chi^2=\sum_{ij}\frac{(O_{ij}-E_{ij})^2}{E_{ij}}\) and chi-squared distribution. |
Are the row and column variables independent? Are the populations homogeneous? data = rbind(TrtA = c(Y = 40, N = 45),TrtB = c(Y = 50, N = 55))chisq.test(data, correct = FALSE)
|
|
Fisher’s
Exact Contingency Table |
The 2x2 cell values and hypergeometric distribution. |
Are the row and column variables independent?new = c(off = 7, on = 12)old = c(off = 12, on = 15)fisher.test(rbind(new, old),alternative = "two.sided",conf.level = 0.95)
|
This section has non-parametric tests and intervals. See the page on each test for more details.
| Test | Formula | Example |
|---|---|---|
|
Sign Tests for One Sample |
\(S = \sum_{i=1}^n
I_{(x_i-m_0)>0}\), & binomial distribution. |
Is the median equal to \(m_0\)?install.packages("BSDA")library(BSDA)data = c(1.4, 0.6, -0.5, 1.7, -1.3, 0.6)SIGN.test(data, alternative = "two.sided",md = 0, conf.level = 0.95)
|
|
Sign Tests for Paired Samples |
\(S = \sum_{i=1}^n
I_{([x_i-y_i]-m_0)>0}\), & binomial distribution. |
Is the median of paired x and y differences equal to \(m_0\)? install.packages("BSDA")library(BSDA)data_x = c(5.4, 8.1, 6.9, 6.4, 5.7)data_y = c(7.7, 6.1, 9.4, 7.0, 5.7)SIGN.test(data_x, data_y,alternative = "two.sided",md = 0, conf.level = 0.95)
|
|
Wilcoxon
Rank-Sum (Mann–Whitney U) Tests |
\(R_i\) is the rank of \(x_i - m_0\), \(W = \sum_{i=1}^{n_x} R_i - \frac{n_x(n_x+1)}{2}\), \(z = \frac{W - \frac{n_x n_y}{2}}{\sqrt{\frac{n_x n_y}{12}\left((n + 1) - \frac{\sum_{k=1}^{T}(t_k^3-t_k)}{n(n-1)} \right)}}\) and standard normal distribution. |
Are the medians equal, or difference equal to \(m_0\)?data_x = c(4.7, 4.1, 4.2, 3.1, 3.8)data_y = c(4.5, 3.7, 5.1, 5.7,3.9, 5.0, 4.6, 4.3)wilcox.test(data_x, data_y,mu = 0, alternative = "two.sided",conf.int = TRUE, conf.level = 0.95)
|
|
Wilcoxon
Signed-Rank Tests for One Sample |
\(R_i\) is the rank of \(|x_i - m_0|\), \(V = \sum_{i=1}^N R_i \cdot I_{(x_i-m_0)>0}\), with \(t_k\) as the number of tied values for set \(k\) that are tied at a particular value, \(z = \frac{V - \frac{n(n+1)}{4}}{\sqrt{\frac{n(n + 1)(2n + 1)}{24}-\frac{\sum_{k=1}^{T}(t_k^3-t_k)}{48}}}\) and standard normal distribution. |
Is the median equal to \(m_0\)?data = c(0.3, 0.5, -1.1, -0.6, 1.3, 0.4)wilcox.test(data, alternative = "two.sided",mu = 0,conf.int = TRUE, conf.level = 0.95)
|
|
Wilcoxon
Signed-Rank Tests for Paired Samples |
\(R_i\) is the rank of \(|(x_i - y_i) - m_0|\), \(V = \sum_{i=1}^N R_i \cdot I_{([x_i-y_i]-m_0)>0}\), with \(t_k\) as the number of tied values for set \(k\) that are tied at a particular value, \(z = \frac{V - \frac{n(n+1)}{4}}{\sqrt{\frac{n(n + 1)(2n + 1)}{24}-\frac{\sum_{k=1}^{T}(t_k^3-t_k)}{48}}}\) and standard normal distribution. |
Is the median of paired x and y differences equal to \(m_0\)? data_x = c(4.1, 4.5, 3.8, 5.2, 2.6)data_y = c(2.6, 3.3, 2.4, 4.8, 6.0)wilcox.test(data_x, data_y,alternative = "two.sided",mu = 0, paired = TRUE,conf.int = TRUE, conf.level = 0.95)
|
|
Spearman’s
Rank Correlation Coefficient Tests |
\(r(x_i)\) and \(r(y_i)\) are ranks of \(x_i\) and \(y_i\), \(r_s = \frac{\sum ^n _{i=1}[r(x_i) - \bar r(x)][r(y_i) - \bar r(y)]}{\sqrt{\sum ^n _{i=1}[r(x_i) - \bar r(x)]^2} \sqrt{\sum ^n _{i=1}[r(y_i) - \bar r(y)]^2}}\), \(t = \frac{r_s}{\sqrt{\frac{1 - r_s^2}{n-2}}}\) and Student’s t-distribution. |
Is there a rank correlation between x and y?data_x = c(3.5, 2.4, 2.8, 2.1, 1.8)data_y = c(6.9, 7.2, 8.2, 7.3, 6.8)cor.test(data_x, data_y,alternative = "two.sided",method = "spearman")
|
| Kruskal-Wallis Tests |
\(\bar r_{j\cdot}\) is the average rank
for group \(j\), \(t_i\) is the number of tied values for set \(i\) that are tied at a particular value, \(H = \frac{\frac{12}{N(N+1)} \sum_{j=1}^G n_j \bar r_{j\cdot}^2 - 3(N+1)}{1-\frac{\sum_{i=1}^T (t_i^3-t_i)}{N^3-N}}\) and chi-squared distribution. |
Are the group medians equal?x = c(17.9, 20.3, 20.4, 20.8, 19.2,17.8, 22.6, 19.2, 18.4, 19.4, 19.3,22.1, 20.3, 21.4, 19.1)groups = c("A", "A", "A", "A", "A","B", "B", "B", "B", "B", "B","C", "C", "C", "C")kruskal.test(x ~ groups)
|
| Kolmogorov-Smirnov Tests |
\(D_n= \sup_x |F_n(x)-F(x)|\) \(D_{n,m}=\sup_x |F_{1,n}(x)-F_{2,m}(x)|\) |
Is the sample from the specified distribution? Are the samples from the same distribution? sample1 = c(4.1, 5.9, 5.2, 3.9, 3.6, 7.1, 4.3)sample2 = c(3.1, 5.5, 5.3, 3.6, 3.1, 4.9, 6.1)ks.test(sample1, pnorm, mean = 5, sd = 1)ks.test(sample1, sample2)
|
| Tests for Normal Distribution |
Histogram Quantile-Quantile plot Shapiro-Wilk test Kolmogorov-Smirnov test |
Is the sample from a normal distribution?sample = c(4.6, 3.8, 6.7, 5.0, 6.1, 2.4,4.5, 4.3, 3.9, 6.5, 3.6, 5.3, 5.8, 5.2)hist(sample)qqnorm(sample); qqline(sample)shapiro.test(sample)ks.test(sample, pnorm,mean(sample), sd(sample))
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