Regression Approach To ANOVA

Slides:



Advertisements
Similar presentations
1-Way Analysis of Variance
Advertisements

Experimental Statistics - week 5
Topic 12: Multiple Linear Regression
1 Chapter 4 Experiments with Blocking Factors The Randomized Complete Block Design Nuisance factor: a design factor that probably has an effect.
Design of Experiments and Analysis of Variance
ANOVA: Analysis of Variation
Part I – MULTIVARIATE ANALYSIS
PSYC512: Research Methods PSYC512: Research Methods Lecture 19 Brian P. Dyre University of Idaho.
Lecture 9: One Way ANOVA Between Subjects
Wrap-up and Review Wrap-up and Review PSY440 July 8, 2008.
Analysis of Variance & Multivariate Analysis of Variance
January 7, morning session 1 Statistics Micro Mini Multi-factor ANOVA January 5-9, 2008 Beth Ayers.
Outline Single-factor ANOVA Two-factor ANOVA Three-factor ANOVA
Experimental Design and the Analysis of Variance.
Linear Regression/Correlation
Linear Regression and Correlation Explanatory and Response Variables are Numeric Relationship between the mean of the response variable and the level of.
T WO W AY ANOVA W ITH R EPLICATION  Also called a Factorial Experiment.  Factorial Experiment is used to evaluate 2 or more factors simultaneously. 
T WO WAY ANOVA WITH REPLICATION  Also called a Factorial Experiment.  Replication means an independent repeat of each factor combination.  The purpose.
Inferential Statistics
QNT 531 Advanced Problems in Statistics and Research Methods
INFERENTIAL STATISTICS: Analysis Of Variance ANOVA
230 Jeopardy Unit 4 Chi-Square Repeated- Measures ANOVA Factorial Design Factorial ANOVA Correlation $100 $200$200 $300 $500 $400 $300 $400 $300 $400 $500.
The following Analysis of Variance table lists the results from a two-factor experiment. Factor A was whether shelf price was raised or not, and factor.
BIOL 4605/7220 Ch 13.3 Paired t-test GPT Lectures Cailin Xu October 26, 2011.
Chapter 12: Introduction to Analysis of Variance
B AD 6243: Applied Univariate Statistics Repeated Measures ANOVA Professor Laku Chidambaram Price College of Business University of Oklahoma.
Chapter 10 Analysis of Variance.
ANOVA (Analysis of Variance) by Aziza Munir
1 G Lect 14a G Lecture 14a Examples of repeated measures A simple example: One group measured twice The general mixed model Independence.
Chapter 14 – 1 Chapter 14: Analysis of Variance Understanding Analysis of Variance The Structure of Hypothesis Testing with ANOVA Decomposition of SST.
ANOVA: Analysis of Variance. The basic ANOVA situation Two variables: 1 Nominal, 1 Quantitative Main Question: Do the (means of) the quantitative variables.
Lecture 9-1 Analysis of Variance
Analysis of Covariance Combines linear regression and ANOVA Can be used to compare g treatments, after controlling for quantitative factor believed to.
VI. Regression Analysis A. Simple Linear Regression 1. Scatter Plots Regression analysis is best taught via an example. Pencil lead is a ceramic material.
1 ANALYSIS OF VARIANCE (ANOVA) Heibatollah Baghi, and Mastee Badii.
Chapter Seventeen. Figure 17.1 Relationship of Hypothesis Testing Related to Differences to the Previous Chapter and the Marketing Research Process Focus.
Analysis of RT distributions with R Emil Ratko-Dehnert WS 2010/ 2011 Session 07 –
1 Overview of Experimental Design. 2 3 Examples of Experimental Designs.
Inferential Statistics. The Logic of Inferential Statistics Makes inferences about a population from a sample Makes inferences about a population from.
Single-Factor Studies KNNL – Chapter 16. Single-Factor Models Independent Variable can be qualitative or quantitative If Quantitative, we typically assume.
Categorical Independent Variables STA302 Fall 2013.
Text Exercise 12.2 (a) (b) (c) Construct the completed ANOVA table below. Answer this part by indicating what the f test statistic value is, what the appropriate.
SPSS meets SPM All about Analysis of Variance
IE241: Introduction to Design of Experiments. Last term we talked about testing the difference between two independent means. For means from a normal.
Comparing k > 2 Groups - Numeric Responses Extension of Methods used to Compare 2 Groups Parallel Groups and Crossover Designs Normal and non-normal data.
Experimental Design and the Analysis of Variance.
Other experimental designs Randomized Block design Repeated Measures designs.
Introduction to ANOVA Research Designs for ANOVAs Type I Error and Multiple Hypothesis Tests The Logic of ANOVA ANOVA vocabulary, notation, and formulas.
Chapter 9 More Complicated Experimental Designs. Randomized Block Design (RBD) t > 2 Treatments (groups) to be compared b Blocks of homogeneous units.
Topic 27: Strategies of Analysis. Outline Strategy for analysis of two-way studies –Interaction is not significant –Interaction is significant What if.
Comparing I > 2 Groups - Numeric Responses Extension of Methods used to Compare 2 Groups Independent and Dependent Samples Normal and non-normal data structures.
Analysis of Variance Yonghui Dong 03/12/2013. Why not multiple t tests comparison? Three Groups comparison: Group 1 vs. Group 2 Group 1 vs. Group 3 Group.
STA248 week 121 Bootstrap Test for Pairs of Means of a Non-Normal Population – small samples Suppose X 1, …, X n are iid from some distribution independent.
CHAPTER 15: THE NUTS AND BOLTS OF USING STATISTICS.
Chapter 11 Analysis of Variance
ANOVA: Analysis of Variation
Two way ANOVA with replication
i) Two way ANOVA without replication
Two way ANOVA with replication
More Complicated Experimental Designs
Linear Contrasts and Multiple Comparisons (§ 8.6)
Comparing Several Means: ANOVA
Chapter 11 Analysis of Variance
Single-Factor Studies
Single-Factor Studies
More Complicated Experimental Designs
More Complicated Experimental Designs
ANOVA: Analysis of Variance
Presentation transcript:

Regression Approach To ANOVA Dummy (Indicator) Variables: Variables that take on the value 1 if observation comes from a particular group, 0 if not. If there are I groups, we create I-1 dummy variables. Individuals in the “baseline” group receive 0 for all dummy variables. Statistical software packages typically assign the “last” (Ith) category as the baseline group Statistical Model: E(Y) = b0 + b1Z1+ ... + bI-1ZI-1 Zi =1 if observation is from group i, 0 otherwise Mean for group i (i=1,...,I-1): mi = b0 + bi Mean for group I: mI = b0

2-Way ANOVA 2 nominal or ordinal factors are believed to be related to a quantitative response Additive Effects: The effects of the levels of each factor do not depend on the levels of the other factor. Interaction: The effects of levels of each factor depend on the levels of the other factor Notation: mij is the mean response when factor A is at level i and Factor B at j

Example - Thalidomide for AIDS Response: 28-day weight gain in AIDS patients Factor A: Drug: Thalidomide/Placebo Factor B: TB Status of Patient: TB+/TB- Subjects: 32 patients (16 TB+ and 16 TB-). Random assignment of 8 from each group to each drug). Data: Thalidomide/TB+: 9,6,4.5,2,2.5,3,1,1.5 Thalidomide/TB-: 2.5,3.5,4,1,0.5,4,1.5,2 Placebo/TB+: 0,1,-1,-2,-3,-3,0.5,-2.5 Placebo/TB-: -0.5,0,2.5,0.5,-1.5,0,1,3.5

ANOVA Approach Total Variation (SST) is partitioned into 4 components: Factor A: Variation in means among levels of A Factor B: Variation in means among levels of B Interaction: Variation in means among combinations of levels of A and B that are not due to A or B alone Error: Variation among subjects within the same combinations of levels of A and B (Within SS)

ANOVA Calculations Balanced Data (n observations per treatment)

ANOVA Approach General Notation: Factor A has I levels, B has J levels Procedure: Test H0: No interaction based on the FAB statistic If the interaction test is not significant, test for Factor A and B effects based on the FA and FB statistics

Example - Thalidomide for AIDS Individual Patients Group Means

Example - Thalidomide for AIDS There is a significant Drug*TB interaction (FDT=5.897, P=.022) The Drug effect depends on TB status (and vice versa)

Regression Approach General Procedure: Generate I-1 dummy variables for factor A (A1,...,AI-1) Generate J-1 dummy variables for factor B (B1,...,BJ-1) Additive (No interaction) model: Tests based on fitting full and reduced models.

Example - Thalidomide for AIDS Factor A: Drug with I=2 levels: D=1 if Thalidomide, 0 if Placebo Factor B: TB with J=2 levels: T=1 if Positive, 0 if Negative Additive Model: Population Means: Thalidomide/TB+: b0+b1+b2 Thalidomide/TB-: b0+b1 Placebo/TB+: b0+b2 Placebo/TB-: b0 Thalidomide (vs Placebo Effect) Among TB+/TB- Patients: TB+: (b0+b1+b2)-(b0+b2) = b1 TB-: (b0+b1)- b0 = b1

Example - Thalidomide for AIDS Testing for a Thalidomide effect on weight gain: H0: b1 = 0 vs HA: b1  0 (t-test, since I-1=1) Testing for a TB+ effect on weight gain: H0: b2 = 0 vs HA: b2  0 (t-test, since J-1=1) SPSS Output: (Thalidomide has positive effect, TB None)

Regression with Interaction Model with interaction (A has I levels, B has J): Includes I-1 dummy variables for factor A main effects Includes J-1 dummy variables for factor B main effects Includes (I-1)(J-1) cross-products of factor A and B dummy variables Model: As with the ANOVA approach, we can partition the variation to that attributable to Factor A, Factor B, and their interaction

Example - Thalidomide for AIDS Model with interaction:E(Y)=b0+b1D+b2T+b3(DT) Means by Group: Thalidomide/TB+: b0+b1+b2+b3 Thalidomide/TB-: b0+b1 Placebo/TB+: b0+b2 Placebo/TB-: b0 Thalidomide (vs Placebo Effect) Among TB+ Patients: (b0+b1+b2+b3)-(b0+b2) = b1+b3 Thalidomide (vs Placebo Effect) Among TB- Patients: (b0+b1)-b0= b1 Thalidomide effect is same in both TB groups if b3=0

Example - Thalidomide for AIDS SPSS Output from Multiple Regression: We conclude there is a Drug*TB interaction (t=2.428, p=.022). Compare this with the results from the two factor ANOVA table

Repeated Measures ANOVA Goal: compare g treatments over t time periods Randomly assign subjects to treatments (Between Subjects factor) Observe each subject at each time period (Within Subjects factor) Observe whether treatment effects differ over time (interaction, Within Subjects)

Repeated Measures ANOVA Suppose there are N subjects, with ni in the ith treatment group. Sources of variation: Treatments (g-1 df) Subjects within treatments aka Error1 (N-g df) Time Periods (t-1 df) Time x Trt Interaction ((g-1)(t-1) df) Error2 ((N-g)(t-1) df)

Repeated Measures ANOVA To Compare pairs of treatment means (assuming no time by treatment interaction, otherwise they must be done within time periods and replace tn with just n):