Inferential Statistics

Descriptive statistics

• methods and procedures for summarizing and describing data

  • visually and numerically

  • ex: mean, median, SD, %

Inferential Statistics

• makes generalizations about a population based on info gathered from samples drawn from that population

• ex: student’s t test, ANOVA, chi-square

Research hypothesis:

specific prediction of the relationship b/w 2 or more sets of data or an outcome of interest

Statistical Hypothesis

• assumption about a population parameter evaluated by statistical techniques

  • null hypothesis: needs to be “nullified”

  • alternative hypothesis: chosen if evidence leads to rejection of null hypothesis

“Does taking ASA reduce the risk of CVD”

What is the research hypothesis?

The risk of CVD is lower compared to those not taking ASA

“Does taking ASA reduce the risk of CVD”

What is the null hypothesis?

no difference in risk of CVD b/w ASA users and non-ASA users

“Does taking ASA reduce the risk of CVD”

What is the alternative hypothesis?

There is a difference in risk of CVD b/w ASA users and non-ASA users

Hypothesis test

method to decide between 2 competing claims to infer information about a population parameter

Describe the steps in hypothesis testing

1. identify the null and alternative hypothesis

2. Consider the assumptions: normal distribution, independent observations

3. set alpha = 5%

4. draw a sample from population of interest

5. calculate test stats

6. draw conclusions based on p-value:

   • reject null hypothesis OR

   • fail to reject the null hypothesis

Parametric Distribution

• data has an underlying distribution that or normal or close to being normal

  • symmetric unimodal curve

Non-parametric distribution

• no assumptions about the data

• unable to understand how far apart the data is

• fixed by ranking the data

Parametric stats

• estimate parameters

• good for continuous data (interval/ratio)

• normal data

• large sample size

• only for parametric data

Non-parametric stats

• does not estimate parameters

• good for nominal/ordinal data

• no assumptions

• large sample size NOT required

• continuous data/non-parametric

Statistical Inference techniques for continuous data

• parametric tests

  • independent t test

  • paired t test

  • ANOVA

• non-parametric tests (skewed/ordinal data)

  • mann-whitney U (wilcoxon’s rank-sum test)

  • wilcoxon’s signed rank test

  • Kruskal-Wallis test

Independent (unpaired) t-test

• used to test differences in mean of a variable b/w 2 independent groups

  • tests null hypothesis that 2 independent populations have the same mean

Assumptions of the independent t-test

• variable is normally distributed

• variances are equal in 2 populations

• observations are independent

Paired (dependent) t-test

• used when the independent assumption is violated

• tests null hypothesis that the 2 means of a dependent population are the same

ANOVA

• when there are more than 2 independent groups

• tests null hypothesis that >/= independent populations have the same mean

• significant result does not tell you which mean is different

Mann-whitney U (Wilcoxon’s Rank-sum) test

• studies differences b/w 2 independent groups on an outcome variable that is ordinal or continuous

  • non-parametric version of independent t test

• tests null hypothesis that there is no difference in the sum of ranks b/w 2 groups

Wilcoxon’s signed rank test

• studies differences b/w 2 dependent groups on an outcome variable that is ordinal/continuous

  • non-parametric version of paired t-test

• no math calculations

• tests null hypothesis that the # of + signs = # of - signs

Kruskal-Wallis Test

• studies differences b/w 3 or more independent groups on an outcome variable that is ordinal/continuous

• non-parametric version of ANOVA

• extension of wilcoxon’s rank-sum test

• tests null hypothesis that there is no difference in the sum of ranks among each group

Statistical inference techniques for categorical (nominal) data

• Chi-squae

• fisher’s exact test

• McNemar’s test

T/F: All tests for categorical data are non-parametric tests.

true

Chi-square test

• determines whether 2 or more proportions are different from one another

• tests null hypothesis that there is no association b/w 2 categorical variables

Assumptions for chi-square test

• groups are independent

• large sample size

Fisher’s Exact Test

• used when sample sizes are not sufficiently large

• tests null hypothesis that there is no association between categorical variables

• groups are independent

McNemar’s test

• tests differences in proportions for paired (dependent) data or correlated samples

Correlation

• correlation coefficient provides index that reflects a quantitative measure of relationship b/w 2 variables

No correlation

0

Weak correlation

0.2-0.4

Moderate correlation

0.4-0.7

Strong correlation

0.7-0.9

Perfect correlation

1

What is the limitation of correlations?

does not establish causality

Pearson correlation coefficient

• for parametric data → linear associations b/w 2 continuous variables

  • when both X and Y are interval/ratio scale data with normal distribution

What is the limitation of pearson correlation coefficient?

very sensitive to outlier data

What is the spearman rho?

• correlation for non parametric data

• for ranked/ordinal data, non-normal ratio data

• reduces influence of outlier data

• linear/non-linear

• monotonic relationship b/w variables

What is regression?

• prediction model

• describes the relationship of one variable with another variable

  • magnitude of change b/w 2 variables