Summary on HT

Disclaimer

! REMEMBER:

• When we fail to reject H0, this means, that due to p ≥ 0.05 there is not enough evidence to reject H0 (so we do never believe it’s true). But if we fail to reject H0 (due to p < 0.05), it means that we can assume H1 to be true.
• if the sample is too small(ca.<30 obs) or too big(ca.>400), always check the normality of distribution by qq-plot
• if distribution isn’t normal, then non-parametric test
• There is one pair of hypotheses to every p-value estimated
• for manual Bonferroni correction multiply a p-value by the number of pairs given in the test
• Regarding one variable of three or more independent samples. We firstly check normality of the variable. And so, if this assumption isn’t met, we don’t consider the assumption of homoscedasticity. We get straight to the non-parametric test
• Attention: in tests related to population proportion it is common that the test is one sided.

0. Relevant for every topic

0.1. Shapiro-Wilk normality test

• Hypotheses
  • H0: the {variable} is normally distributed
  • H1: the {variable} isn’t normally distributed

1. ONE POPULATION BEING STUDIED

1.1.The one sample (parametric) t-test [mean, sd, etc]

• Assumptions
  • Numeric variable
  • Normal distribution of VARIABLE (to check by histogram, Shapiro-Wilk normality test, and ggqqplot, if needed)
• Hypotheses
  • H0:µ=… [the mean value of {variable} equals…]
  • H1:µ≠… [the mean value of {variable} is not equal to…]
• Effect size: CohenD (Sawilowsky)

1.2.The Wilcoxon signed rank (non-parametric) test [median or else]

• Assumptions
  • Numeric variable
  • No normal distribution of variable
• Hypotheses
  • H0:Me = … [the median of {variable} equals …]
  • H1:Me ≠ … [the median of {variable} is not equal to …]
• Effect size: Bisserial correlation (Funder)

2. ONE POPULATION OBSERVED TWICE (two variables)

2.1.Paired (parametric) t-test [mean etc]

• Assumptions
  • Numeric Variable
  • Normal distribution of DIFFERENCES between values of two variables (to check by histogram, Shapiro-Wilk normality test, and ggqqplot, if needed)
• Hypotheses
  • H0:µd = 0 (µa = µb) [the mean difference is equal to 0]
  • H1:µd ≠ 0 (µa ≠ µb) [the mean difference isn’t equal to 0]
• Effect size: CohenD (Sawilowsky)

2.2.The Wilcoxon signed rank (non-parametric) test [f.e.median]

• Assumptions
  • Numeric variable
  • No normal distribution of differences
• Hypotheses
  • H0: the distribution locations of both variables are same
  • H1: the distribution locations of both variables aren’t same
• Effect size: Bisserial correlation (Funder)

3. ONE VARIABLE OF TWO INDEPENDENT POPULATIONS

3.1.Independent samples (parametric) t-test

• Assumptions
  • Data from two different populations
  • Numeric variable
  • Normal distribution of VARIABLE IN BOTH POPULATIONS (to check by histogram, Shapiro-Wilk (grouped through pipe operators) normality test, and ggqqplot, if needed)
  • Same (equal) variance of variable in both populations (to check by levenetest).

!If variances of the variable in both samples aren’t same, then apply Welch correction, writing an argument ‘var.equal = FALSE’ in the middle of the independent samples t-test

• Hypotheses
  • H0:µ1-µ2 = 0 (µ1pop = µ2pop) [the mean values of the {variable} in both groups are same]
  • H1:µ1-µ2 ≠ 0 (µ1pop ≠ µ2pop) [the mean values of the {variable} in both groups are different]
• Effect size: CohenD (Sawilowsky)

3.2. The Wilcoxon ranked sum (non-parametric) test

• Assumptions
  • Numeric variable
  • No normal distribution of variable in at least one population
• Hypotheses
  • H0: the distribution locations of {variable} for each population are same
  • H1: the distribution locations of {variable} for each population aren’t same
• Effect size: Bisserial correlation (Funder)

3.1.2. Levene Test

• Hypotheses
  • H0: sigma^2a = sigma^2b
  • H1: sigma^2a ≠ sigma^2b

4. ONE VARIABLE OF THREE OR MORE INDEPENDENT SAMPLES

4.1. One-way ANOVA (parametric)

• Assumptions
  • Numeric variable
  • Normal distribution of variable in each group (to check by boxplot, Shapiro-Wilk (grouped through pipe operators) normality test, and ggqqplot, if needed)
  • the numeric variable must have same variance in each group(homogeneity of variance…. it is homoskedastisity…)

!If variances of the variable in each sample aren’t same, then Welch heteroskedastic (parametric) F-test instead of one-way ANOVA.

• Hypotheses
  • H0: µA = µB = µC [the mean values of the {variable} in every group are same]
  • H1: at least one of mean values is different
• Effect size: Eta-squared (Cohen 1992)

!If the H0 was rejected, then we continue the analysis by performing POST-HOC pairwise t-test

4.2. Kruskal-Wallis rank sum (non-parametric) test

• Assumptions
  • Numeric variable
  • No normal distribution of variable in at least one group
• Hypotheses
  • H0: the distribution locations of {variable} among groups are same
  • H1: at least one group’s distribution location is different
• Effect size: function for kruskal effect size

!If the H0 was rejected, when we continue the analysis by performing POST-HOC Wilcoxon rank sum test(check near pairwise t-test)

4.1.1. Levene Test

• Hypotheses
  • H0: 𝜎^2𝐴=𝜎2𝐵=𝜎^2𝐶 [the variance of the {variable} in every group is same]
  • H1: at least one 𝜎^2 is different

4.1.2. Welch heteroskedastic (parametric) F-test

• Hypotheses
  • H0: µA = µB = µC [the mean values of the {variable} in every group are same]
  • H1: at least one of mean values is different
• Effect size: Eta-squared (Cohen 1992)

Post-hoc pairwise t-test

• Hypotheses
  • H0: ∆µ = 0 [There is no statistical difference in µ of the pair]
  • H1: ∆µ ≠ 0 [There is statistical difference in µ of the pair]

Post-hoc Wilcoxon rank sum test

• Hypotheses (ATTENTION: Check adjusted p-values)
  • H0: the distribution locations of the {variable} in both groups are the same
  • H1: the distribution locations of the {variable} in both groups aren’t the same

5. HYPOTHESIS ABOUT POPULATION PROPORTION

5.1. Proportion (parametric) test

• Assumptions
  • nπ > 5 and n(1-π) >5

!If this assumption isn’t met, then binomial test. This assumption actually only makes sure our sample is big enough

• Hypotheses
  • H0: π = … [proportion is equal to …]
  • H1: π ≠ … [proportion is not equal to …]

5.2. Binomial (non-parametric) test

• Assumptions
  • the assumption for the parametric test is violated
• Hypotheses
  • H0: π = … [proportion is equal to …]
  • H1: π ≠ … [proportion is not equal to …]

Hypothesis about the equality of two proportions

• Hypotheses
  • H0: π2 = π2 [proportions are equal]
  • H1: π ≠ π2 [proportion aren’t equal]