Test of Hypothesis Cheatsheet by Chinedu Eleh

Independent t-Test

used on: continuous data assumptions: independent, normally distributed with constant variance used for: comparing the means of two population groups null: \(H_0: \mu_1 = \mu_2 \); statistic: \(t_0 = \frac{\bar{x}_1 - \bar{x}_2}{ \sqrt{S^2_1/N_1 + S^2_2/N_2} } \), reject if \(|t_0|> t_{1-\alpha/2, df} \)

Paired t-Test

used on: continuous data assumptions:normally distributed paired differences with constant variance used for: comparing the performance of a group of subjects under two different test conditioins null: \(H_0: \mu_1 - \mu_2 = 0 \); statistic: \(t_0 = \frac{\bar{d}_1}{ S_d/\sqrt{n} } \), reject if \(|t_0|> t_{1-\alpha/2, n-1} \)

One-Way ANOVA with Repeated Measures

used on: continuous data assumptions: normal and with constant variance in time used for: comparing the means of repeated measurements on the same subjects null: \(H_0: \mu_1 = \mu_2 = \cdots = \mu_k \); statistic: \(F_0 = \frac{MS_{time}}{MS_{error}} \), reject if \(|F_0|> F_{a-1, a(n-1)}(\alpha) \)

Wilcoxon Signed Rank Test

used on: ordinal or continuous data assumptions: independent, normality assumptions not required used for: nonparametric comparison of mean ranks of paired measurements null: \(H_0: R_1 - R_2 = 0\); statistic: \(w_0 = \sum^{N_r}_{r=1} (\text{sgn}(x_{2i} - x_{1i}) R_i) \), reject if \(|w_0| > w_{1-\alpha/2, N_r} \) where \(N_r\) is the number of nozero differences in rank

Friedman Test

used on: ordinal or continuous data assumptions: measurement on at least 3 occasions or measures under different condtions used for: non-parametric comparison of distributions of repeated measurements statistic: \(Q_0 = \frac{12n}{k(k+1)} \sum^k_{j=1} \bigg(\bar{r}_{\cdot j} - \frac{k+1}{2} \bigg)^2 \) where \(\bar{r}_{\cdot j} = \frac{1}{n}\sum^k_{i=1}r_{ij} \) and \( [r_{ij}]_{n\times k} \) is the matrix of ranks corresponding to the data matrix \( [x_{ij}]_{n\times k} \) with \(n\) (the blocks) and \(k\) columns (the treatments)

One-Way ANOVA

used on: continuous data assumptions: independent, normally distributed with constant variance used for: comparing the means of population groups null: \(H_0: \mu_1 = \mu_2 = \cdots = \mu_k \); statistic: \(F_0 = \frac{MS_{between}}{MS_{error}} \), reject if \(|F_0|> F_{a-1, a(n-1)}(\alpha) \)

Kruskal-Wallis Test

used on: ordinal or continuous data assumptions: independent not necessarily normal observations with similar group sample sizes used for: nonparametric comparison distributions of groups of measurements null: \(H_0:\) groups have same distribution; statistic: a bit long

Fisher F-Test

used on: continuous data assumptions: independent, normally ditributed groups used for: checking homoscedasticity (homogeneity of variances of two populations) null: \(H_0: \sigma_1^2 = \sigma_2^2\); statistic: \(F_0 = \frac{S^2_1}{S^2_2} \sim F_{n_1-1,n_2-1}\) distribution where \(S_1^2\) is the bigger of the two

Two-Way ANOVA

used on: continuous data assumptions independent, normally distributed groups with constant variance used for: comparing population means for combinations of two factors null: maybe one of three: (1)the population means of the first factor are equal (more like one-way ANOVA for the row factor) (2)the population means of the second factor are equal (more like one-way ANOVA for the column factor) (3)there is no interaction between the two factors statistic:for each hypotheses given above, there is an F-test which is the mean square for each main effect and the interaction effect divided by the within variance

Chi-square Test

used on: categorical data used for: checking if distributions of categorical variables are independent of each other assumptions: no single population parameter, so invloves non-parametric statistic null: all pairs of groups are independent; statistic: \(X^2_0 = \sum^k_{i=1} \frac{(x_i-m_i)^2}{m_i} \sim \chi^2_{k-1}\) distribution where \(k\) is the number of categories, \(x_i, \ m_i\) are the frequency and expectation of \(i\)th class respectively

Proportions Test

used on:ordinal data assumptions:no single population parameter, so invloves non-parametric statistic used for: comparing two populations proportions null: \(H_0: p_1 = p_2\) statistic:may either be odds ratio, relative risk or even chi-square statistcs

Mann-Whitney Test

used on: ordinal or continuous data assumptions: independent, normally distributed with constant variance used for: nonparametric comparison of mean/median ranks of independent measurements null: \(H_0:\) two samples from the same population have the same distribution; statistic: \(u_0 = \min (u_1, u_2) \) where \(u_i = R_i - \frac{n_i(n_i+1)}{2}, \ i=1,2 \) and \(R_i\) is the \(i\)th sample.

Bartlett Test

used on: continuous data assumptions: independent, normally ditributed groups used for: checking homoscedasticity (homogeneity of variances) null: \(H_0: \sigma_1^2 = \sigma_2^2 = \cdots = \sigma_k^2 \); statistic: really long but has approximately a \(\chi^2_{k-1}\) distribution

Levene Test

used on: continuous data assumptions: independent, normally ditributed groups used for: checking homoscedasticity (homogeneity of variances) null: \(H_0: \sigma_1^2 = \sigma_2^2 = \cdots = \sigma_k^2 \); statistic: a bit long but has approximately an \(F_{k-1, N-k}\) distribution and has the characteristic of being less sensitive to normality in contrast to Bartlett Test