THE ASSOCIATION BETWEEN TWO CATEGORICAL VARIABLESWhen analysing data the first step is to distinguish between the response variable and the explanatory variable. The response variable is the outcome variable on which comparisons are made. If the explanatory variable is categorical, it defines the groups to be compared with respect to values for the response variable. If the explanatory variable is quantitative, it defines the change in different numerical values to be compared with respect to values for the response variable. The explanatory variable should explain the response variable (e.g: survival status is a response variable and smoking status is the explanatory variable).An association exists between two variables if a particular value for one variable is more likely to occur with certain values of the other variable. A contingency table is a display for two categorical variables. Conditional proportions are proportions which formation is conditional on ‘x’. A conditional proportion should be conditional to something. A conditional proportion is also a percentage. The proportion of the totals (e.g: percentage of total amount of ‘no’) is called a marginal proportion.There is probably an association between two variables if there is a clear explanatory/response relationship, that dictates which way we compute the conditional proportions. Conditional proportions are useful in determining if there’s an association. A variable can be independent from another variable. THE ASSOCIATION BETWEEN TWO QUANTITATIVE VARIABLESWe examine a scatterplot to study association. There is a difference between a positive association and a negative association. If there is a positive association, x goes up as y goes up. If there is a negative association, x goes up as y goes down. Correlation describes the strength of the linear association. Correlation (r) summarizes th direction of the association between two quantitative variables and the strength of...

ONE-WAY ANOVE: COMPARING SEVERAL MEANSThe inferential method for comparing means of several groups is called analysis of variance, also called ANOVA. Categorical explanatory variables in multiple regression and in ANOVA are referred to as factors, also known as independent variables. An ANOVA with only one independent variable is called a one-way ANOVA. Evidence against the null hypothesis in an ANOVA test is stronger when the variability within each sample is smaller or when the variability between groups is larger. The formula for the F (ANOVA) test is:When the null hypothesis is true, the mean of the F-distribution is approximately 1. If the null hypothesis is wrong, then F>1. This also increases if the sample size increases. The larger the F-statistic, the smaller the P-value. The F-distribution has two degrees of freedom values: and The ANOVA test has five steps:AssumptionsA quantitative response variable for more than two groups. Independent random samples. Normal population distribution with equal standard deviation.HypothesesTest statisticyP-valueThis is the right-tail probability of the observed F-value.ConclusionThe null hypothesis is normally rejected if the P-value is smaller than 0.05.If the sample sizes are equal, the within-groups estimate of the variance is the mean of the g sample variances for the g groups. It uses the following formula: If the sample sizes are equal, the between-groups estimate of the variance uses the following formula: The ANOVA F-test is robust to violations if the sample size is large enough. If the population sample sizes are not equal, the F test works quite well as long as the largest group standard deviation is no more than about twice the smallest group standard deviation. Disadvantages of the F-test are that it tells us whether groups are different, but it does not tell us which groups are different. ESTIMATING DIFFERENCES IN GROUPS FOR A SINGLE FACTORThe F-test only...

Probability refers to the proportion of occurrence when a particular experiment is repeated infinitely often under different circumstances. It is a long-term relative frequency, does not apply to unique events and is dependent on the reference category. Subjective probability refers to the subjective degree of conviction in a hypothesis. Objective probability refers to the long-term relative frequency and is the same probability used in classical statistics. The p-value is the probability of finding a test statistic at least as extreme as the one observed, given that the null hypothesis is true. An X% confidence interval for a parameter is an interval that in repeated use has an X% chance to capture the true value of the parameter. The p-values are only concerned about the null hypothesis, although it is not possible to make statements about the probability of a hypothesis in classical statistics.If the null hypothesis is true, then the p-values drift randomly. Therefore, it is possible that the p-value is significant by chance. This is why stopping rules are imperative in classical statistics. In Bayesian statistics, the Bayes factor does not drift randomly but drifts towards the correct decision.In classical statistics, the stopping rules (1), the timing of explanations (posthoc test or not) (2) and multiple tests influence the conclusion. This is not the case in Bayesian statistics.Classical statistics does not allow for probabilities to be assigned to hypotheses or parameters, whereas Bayesian statistics does allow this.Bayesian statistics is a method of learning from prediction errors. It assumes that probability does not exist but only uncertainty, which has to be quantified in a principled manner. Therefore, in Bayesian statistics, probability can be assigned to a single hypothesis.The data drive an update from prior knowledge to posterior knowledge. This method investigates whereas classical statistics investigates . The Bayes...