Lecture 7: Inferential Statistics

What is inferential statistics used for?

Used to determine if differences (or relationships) exist between groups

What might differences (or relationships) may be due to?

• Sampling error

• True differences in (relationships between) the population means

What do parametric inferential tests uses? What are some examples?

• Use sample statistics such as the mean, standard deviation, and variance 

• Examples:

  • T-tests

  • ANOVAs

  • Certain types of correlation and regression

What does parametric inferential tests assume?

• That the subjects are randomly selected from a normally distributed population

• That if there is more than 1 group -- the variance/variability for each group is the same 

• These tests (e.g., ANOVA, t-tests, etc.) are using interval ratio data

What do non-parametric inferential tests use? What are some examples?

• Use rank, frequency, or median

• Examples:

  • Mann-Whitney

  • Wilcoxon Rank Sum

  • Chi-Square

  • Kruskal-Wallis

  • Certain types of correlation and regression

What do non-parametric inferential tests assume?

• That using nominal / ordinal data

• They they are not normally distributed

What is the independence of measures?

• The ""between-subjects"" analysis

• Groups that are independent from each other

• E.g., looking at differences between two independent groups

What is the dependence of measures?

• Repeated measures within the same group

• The ""within-group"" analysis

What are the Parametric Tests of relationships?

• Pearson Correlation

• Intra-Class Correlation Coefficient

• Linear Regression

What are the Non-Parametric Tests of relationships?

• Spearman Rank

• Kappa Coefficient

• Logistic Regression

What is the purpose of a Pearson Correlation?

What is the relationship between two scale variables (ratio or interval data)

What information is provided by a Pearson Correlation?

• r = correlation coefficient

  • Strength of the association (-1 to +1)

  • Assumes data is linearly related

• Obtained p-value

  • Statistical significance of the association

• Confidence intervals around the r-value

What is the purpose of an Intra-Class Correlation Coefficient?

Reliability of repeated measures

What information provided by Intra-Class Correlation Coefficient?

• ICC = reliability coefficient

  • Strength of the reliability index (-1 to +1)

• Obtained p-value

  • Statistical significance of the coefficient

• Confidence intervals

What is the purpose of Linear Regression?

"To answer the question -- can the outcome be predicted by knowing some known factor (usually independent variable, aka the ""x"" factor)"

What are the two basic types of Linear Regression?

• Simple -- one predictor or factor

• Multiple -- 2 or more predictors or factors

• ""y = a + b₁x₁""

What information is provided by a Simple Regression Analysis?

• r = strength of relationship between the individual predictor (x) and the outcome (y)

• r² = amount of variance in y accounted for by x

• b (beta) = amount of change in y per unit x

• Obtained p-value = the statistical significance of the relationship between x and y

What is the Regression Equation? What does each variable stand for?

y = a + b₁x₁ 

• y = dependent continuous variable (e.g., outcome)

• x = acutal value of independent variable (factor)

• a = constant

• b = regression coefficient (beta); slope of the line

What is the regression equation for a multiple regresion analysis?

y = a + b₁x₁ + b₂x₂ + …

What information does a Multiple Regression Analysis provide?

• Some of the same information as a simple regression:

  • Constant

  • Betas for each x

  • r for each x & y pair

  • Obtained p-value for the equation

• Standardized Beta

  • Analogous to r² 
    

  • How much variance in y is explained by each individual ""x""

  • Tested for significance individually

• R² calculated for entire equation

The Spearman Rank Correlation Coefficient is analogous to ________

Pearson Correlation

What information is provided by the Spearman Rank Correlation Coefficient?

• p (rho)

• p-value

Kappa is analogous to ________

Intraclass Correlation Coefficient

What information is provided by Kappa (k)?

• K

• p-value

Logistic Regression is analogous to ________

Linear Regression

What does Logistic Regression use?

Uses a dichotomous measure of the outcome

What info does a Logistic Regression provide?

Odds ratio (odds of the outcome occurring given a change in the predicotr value)

What two statistics are used to evaluate measurement properties of relationships?

• Intraclass Correlation Coefficient

  • > .75 = excellent

  • < 75 = moderate ot poor

• Cronbach's Alpha

  • 0.70 - 0.90 = ""strong""

  • > 0.90 = redundant items

  • < 0.70 = different traits

What does Chronbach's Alpha measure?

Internal consistency of self-reported instruments (e.g., health status surveys)

What are the Parametric Tests of differences?

• Independent t-Test

• One-Way ANOVA

• Dependent t-Test

• Repeated Measures ANOVa

• *ANCOVA

What are the Non-Parametric Tests of differences?

• Mann-Whitney U

• Kruskal Wallis

• Wilcoxon Signed Rank

• Friedman's ANOVA

What is the purpose of an Independent t-test?

What is the difference between two independent groups

What info is provided by an Independent t-test?

• t-statistic

• p-value

What is the purpose of the One-Way ANOVA?

Is there a difference between two or more groups; one way in this instance means only one independent variable

What information does a One-Way ANOVA provide?

• F-statistic

• p-value

What is required for an ANOVA? Why?

• Post-Hoc Analyses

• This determines where the difference is

What is an ANCOVA?

Essentially same thing as an ANOVA but it factors for extraneous variables

Mann-Whitney U is analogous to ________

Independent T-Test

What information does a Mann-Whitney U provide?

• U-score

• p-value

Kruskal-Wallis is analogous to ________

One-Way ANOVA

What information does the Kruskal Wallis provide?

• H-Statistic!

• Also -- requires a post-HOC test to determine where the difference occurs (just like ANOVA)

What are the Parametric Tests of differences for Dependent Measures?

• Paired T-Test

• Repeated Measures ANOVA

What is the purpose of the paired t-test?

• Is there a difference within a group over time
  

• Compares base-line to post-test

What information does a paired t-test provide?

• T statistic

• p-value

What is the purpose of a Repeated Measures ANOVA?

Differences amongst repeated measures within two or more groups

What information is provided by a Repeated Measures ANOVA?

• F-statistic

• p-value

• (Also -- Post-HOC Analyses is req.)

What are the Non-Parametric Tests of differences for Dependent Measures?

• Wilcoxon Signed Rank Test

• Friedman's ANOVA

Wilcoxon Signed Rank Test is analogous to ________

Paired/Dependent T-Test

For a Wilcoxon Signed Rank Test, what information is provided and when would it be used?

• Used for when there is one independent and one dependent variable that is either nominal or ordinal data

• Provides us with a Z-score

Friedman's ANOVA is analogous to ________

Repeated Measures ANOVA

What information is provided from a Friedman's ANOVA?

• F-statistic

• P-value

• Still need to do post-HOC

What is a Chi-Square Test? When it is typically used and what information does it provide?

• Assesses relative proportions of subjects with a particular characteristic in each group

• Compares observed versus expected frequencies if the null hypothesis (H₀) were true 

• Often used in epidemiology studies (case-control)

• Provides X² and obtained p-value

What is a p-value?

• Evidence against the null hypothesis

• What is the probabiltiy that this difference/relationship si due to chance

• Lower p-value = stronger strength against the null

What is an Alpha Level?

• Level of significance

• Setting this critical level for significance

• Probability of committing a type I error

  • Found a difference but there is no difference -- false positive finding

  • Found a relationship but there is no relationship - false positive finding

What is the Bonferroni Adjustment?

Alpha inflation due to repeated statistical testing

What is a Type I error?

• Concludes that a difference exists when in facter there is no difference 

• ""False Positive"" 
  

• Alpha = probabiltiy of making Type I error

What is a Type II error?

• Conclude that no difference exists when in fact there is a difference

• ""False Negative""
  

• Beta = probability of making a Type II error
  

• Inversely related to alpha

What is Power?

The likelihood that one will detect a difference (relationship) when one exists (1 - Beta)

What are some methods for increasing power?

• Control extraneous variables and apply methods consistently

• Use homogeneous groups

• Increase sample size

What are Sample Size Determinants?

• Desired level of power

• Alpha level to be used in research

• Estimate of effect size (clinically meaningful relationship/difference between groups)