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**Experimental Statistics - week 5**

Chapters 8, 9: Miscellaneous topics Chapter 14: Experimental design concepts Chapter 15: Randomized Complete Block Design (15.3)

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**yij = mi + eij yij = m + ai + eij 1-Factor ANOVA Model or**

observed data unexplained part mean for ith treatment

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were rewritten as:

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In words: TSS(total SS) = total sample variability among yij values SSB(SS “between”) = variability explained by differences in group means SSW(SS “within”) = unexplained variability (within groups)

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**Analysis of Variance Table**

Note: unequal sample sizes allowed

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**Extracted from From Ex. 8.2, page 390-391**

3 Methods for Reducing Hostility 12 students displaying similar hostility were randomly assigned to 3 treatment methods. Scores (HLT) at end of study recorded. Method Method Method Test:

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**ANOVA Table Output – extracted hostility data - calculations done in class**

Source SS df MS F p-value Between <.001 samples Within Totals

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**Fisher’s Least Significant Difference (LSD)**

Protected LSD: Preceded by an F-test for overall significance. Only use the LSD if F is significant. X Unprotected: Not preceded by an F-test (like individual t-tests).

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**Hostility Data - Completely Randomized Design The GLM Procedure **

t Tests (LSD) for score NOTE: This test controls the Type I comparisonwise error rate, not the experimentwise error rate. Alpha Error Degrees of Freedom Error Mean Square Critical Value of t Least Significant Difference Means with the same letter are not significantly different. t Grouping Mean N method A B B B

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**Notice unequal sample sizes**

Ex. 8.2, page 3 Methods for Reducing Hostility 24 students displaying similar hostility were randomly assigned to 3 treatment methods. Scores (HLT) at end of study recorded. Method Method Method Notice unequal sample sizes Test:

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**ANOVA Table Output – full hostility data**

Source SS df MS F p-value Between <.0001 samples Within Totals

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The GLM Procedure t Tests (LSD) for score NOTE: This test controls the Type I comparisonwise error rate, not the experimentwise error rate. Alpha Error Degrees of Freedom Error Mean Square Critical Value of t Comparisons significant at the 0.05 level are indicated by ***. Difference method Between % Confidence Comparison Means Limits *** *** *** *** *** *** Notice the different format since there is not one LSD value with which to make all pairwise comparisons.

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**Duncan's Multiple Range Test for score**

NOTE: This test controls the Type I comparisonwise error rate, not the experimentwise error rate. Alpha Error Degrees of Freedom Error Mean Square Harmonic Mean of Cell Sizes NOTE: Cell sizes are not equal. Number of Means Critical Range Means with the same letter are not significantly different. Duncan Grouping Mean N method A B C Note: Duncan’s test (another multiple comparison test) avoids the issue of different sample sizes by using the harmonic mean of the ni’s.

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**Some Multiple Comparison Techniques in SAS**

FISHER’S LSD (LSD) BONFERONNI (BON) DUNCAN STUDENT-NEWMAN-KEULS (SNK) DUNNETT RYAN-EINOT-GABRIEL-WELCH (REGWQ) SCHEFFE TUKEY

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**Balloon Data Col. 1-2 - observation number**

Col observation number Col color (1=pink, 2=yellow, 3=orange, 4=blue) Col inflation time in seconds 1122.4 2324.6 3120.3 4419.8 5324.3 6222.2 7228.5 8225.7 9320.2

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**Balloon Data Col. 1-2 - observation number**

Col observation number Col color (1=pink, 2=yellow, 3=orange, 4=blue) Col inflation time in seconds 1122.4 2324.6 3120.3 4419.8 5324.3 6222.2 7228.5 8225.7 9320.2

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ANOVA --- Balloon Data General Linear Models Procedure Dependent Variable: TIME Sum of Mean Source DF Squares Square F Value Pr > F Model Error Corrected Total R-Square C.V Root MSE TIME Mean Mean Source DF Type I SS Square F Value Pr > F Color

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**ANOVA --- Balloon Data The GLM Procedure t Tests (LSD) for time**

NOTE: This test controls the Type I comparisonwise error rate, not the experimentwise error rate. Alpha Error Degrees of Freedom Error Mean Square Critical Value of t Least Significant Difference Means with the same letter are not significantly different. t Grouping Mean N color A A A B B B

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**Experimental Design: Concepts and Terminology**

Designed Experiment - an investigation in which a specified framework is used to compare groups or treatments Factors - any feature of the experiment that can be varied from trial to trial - up to this point we’ve only looked at experiments with a single factor

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**- these are called replicates**

Treatments - conditions constructed from the factors (levels of the factor considered, etc.) Experimental Units - subjects, material, etc. to which treatment factors are randomly assigned - there is inherent variability among these units irrespective of the treatment imposed Replication - we usually assign each treatment to several experimental units - these are called replicates

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**Examples: 1. factor 2. treatments 3. experimental units 4. replicates**

Car Data Hostility Data Balloon Data

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**Question: Balloon Data**

1122.4 2324.6 3120.3 4419.8 5324.3 6222.2 7228.5 8225.7 9320.2 Balloon Data Col observation number (run order) Col color (1=pink, 2=yellow, 3=orange, 4=blue) Col inflation time in seconds Question: Why randomize run order? i.e. why not blow up all the pink balloons first, blue balloons next, etc?

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**Scatterplot Using GPLOT What do we learn from this plot?**

Time Run Order What do we learn from this plot?

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**RECALL: 1-Factor ANOVA Model**

- random errors follow a Normal (N) distribution, are independently distributed (ID), and have zero mean and constant variance -- i.e. variability does not change from group to group

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**Checking Validity of Assumptions**

Model Assumptions: - equal variances - normality Checking Validity of Assumptions Equal Variances 1. F-test similar to 2-sample case - Hartley’s test (p.366 text) - not recommended 2. Graphical - side-by-side box plots

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**Graphical Assessment of Equal Variance Assumption**

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**Note: Optional approaches if equal variance assumption is violated:**

1. Use Kruskal Wallis nonparametric procedure – Section 8.6 2. Transform the data to induce more nearly equal variances – Section 8.5 -- log -- square root Note: These transformations may also help induce normality

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**Assessing Normality of Errors The e ij’s are called residuals.**

yij = m + ai + eij so eij = yij - (m + ai) = yij - mi eij is estimated by The e ij’s are called residuals.

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**SAS Code for Balloon Data**

proc glm; class color; model time=color; title 'ANOVA --- Balloon Data'; output out=new r=resball; means color/lsd; run; proc sort; by color; proc boxplot; plot time*color; title 'Side-by-Side Box Plots for Balloon Data'; proc univariate; var resball; histogram resball/normal; title 'Histogram of Residuals -- Balloon Data'; proc univariate normal plot; title 'Normal Probability Plot for Residuals - Balloon Data'; proc gplot; plot time*id; title 'Scatterplot of Time vs ID (Run Order)';

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**Normal Probability Plot**

*+ | * *+++ | *+++ | *+ | *** | **** ***+ | ** | *** | ***** | *+ | *+*+* * ++++

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**Caution: Chapter 15 introduces some new notation**

- i.e. changes notation already defined

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**Recall: Sum-of-Squares Identity 1-Factor ANOVA**

In words: Total SS = SS between samples + within sample SS

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**- new notation for Chapter 15**

Recall: Sum-of-Squares Identity Factor ANOVA - new notation for Chapter 15

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**- new notation for Chapter 15**

Recall: Sum-of-Squares Identity Factor ANOVA - new notation for Chapter 15

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**- new notation for Chapter 15**

Recall: Sum-of-Squares Identity Factor ANOVA - new notation for Chapter 15 In words: Total SS = SS for “treatments” + SS for “error”

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**Revised ANOVA Table for 1-Factor ANOVA (Ch. 15 terminology - p.857)**

Source SS df MS F Treatments SST t - 1 Error SSE N - t Total TSS N - 1

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**yij = mi + eij yij = m + ai + eij**

Recall 1-factor ANOVA (CRD) Model for Gasoline Octane Data yij = mi + eij or yij = m + ai + eij observed octane mean for ith gasoline unexplained part -- car-to-car differences -- temperature -- etc.

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**Similar to diet t-test example:**

Gasoline Octane Data Question: What if car differences are obscuring gasoline differences? Similar to diet t-test example: Recall: person-to-person differences obscured effect of diet

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**Possible Alternative Design for Octane Study:**

Test all 5 gasolines on the same car - in essence we test the gasoline effect directly and remove effect of car-to-car variation Question: How would you randomize an experiment with 4 cars?

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**Blocking an Experiment**

- dividing the observations into groups (called blocks) where the observations in each block are collected under relatively similar conditions - comparisons can many times be made more precisely this way

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**Terminology is based on Agricultural Experiments**

Consider the problem of testing fertilizers on a crop - t fertilizers - n observations on each

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**Completely Randomized Design**

B A A C B B A C A C C B C A t = 3 fertilizers n = 5 replications B - randomly select 15 plots - randomly assign fertilizers to the 15 plots

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**Randomized Complete Block Strategy**

A | C | B B | A | C C | B | A A | B | C t = 3 fertilizers C | A | B - select 5 “blocks” - randomly assign the 3 treatments to each block Note: The 3 “plots” within each block are similar - similar soil type, sun, water, etc

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**Randomized Complete Block Design**

Randomly assign each treatment once to every block Car Example Car 1: randomly assign each gas to this car Car 2: etc. Agricultural Example Randomly assign each fertilizer to one of the 3 plots within each block

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**Model For Randomized Complete Block (RCB) Design**

yij = m + ai + bj + eij effect of ith treatment effect of jth block unexplained error (gasoline) (car) -- temperature -- etc.

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**Previous Data Table from Chapter 8 for 1-factor ANOVA**

column averages don’t make any sense

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**Back to Octane data: “Restructured” Data Car Old Data Format Gas Gas**

Suppose that instead of 20 cars, there were only 4 cars, and we tested each gasoline on each car. “Restructured” Data Car Old Data Format A B C D E A B C D E Gas Gas

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**- using new notation for Chapter 15**

Recall: Sum-of-Squares Identity Factor ANOVA - using new notation for Chapter 15 In words: Total SS = SS for “treatments” + SS for “error”

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**A New Sum-of-Squares Identity**

In words: Total SS = SS for treatments + SS for blocks + SS for error

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**Hypotheses: To test for treatment effects - i.e. gas differences**

we test To test for block effects - i.e. car differences (not usually the research hypothesis) we test

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**Randomized Complete Block Design ANOVA Table**

Source SS df MS F Treatments SST t - 1 Blocks SSB Error SSE Total TSS bt - 1 See page 866

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**Test for Treatment Effects**

Note:

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Test for Block Effects

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**“Restructured” CAR Data - SAS Format**

A B A B A B A B B B B B B B B B C B C B C B C B D B D B D B D B E B E B E B E B The first variable (A - E) indicates gas as it did with the Completely Randomized Design. The second variable (B1 - B4) indicates car.

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**SAS file - Randomized Complete Block Design for CAR Data**

INPUT gas$ block$ octane; PROC GLM; CLASS gas block; MODEL octane=gas block; TITLE 'Gasoline Example -Randomized Complete Block Design'; MEANS gas/LSD; RUN;

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**1-Factor ANOVA Table Output - octane data**

Source SS df MS F p-value Gas (treatments) Error Totals

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**1-Factor ANOVA Table Output - car data**

Source SS df MS F p-value Gas (treatments) Cars (blocks) Error Totals

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**SAS Output -- RCB CAR Data**

Dependent Variable: OCTANE Sum of Mean Source DF Squares Square F Value Pr > F Model Error Corrected Total R-Square C.V Root MSE OCTANE Mean Source DF Anova SS Mean Square F Value Pr > F GAS BLOCK

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**Multiple Comparisons in RCB Analysis**

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**CAR Data -- LSD Results CRD Analysis RCB Analysis **

t Grouping Mean N gas A E B D B C B C C B C B B C C A RCB Analysis t Grouping Mean N gas A E B D B C B C C C B C A

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**CAR Data -- Bonferroni Results**

CRD Analysis Bon Grouping Mean N gas A E A B A D B B C B B B A RCB Analysis Bon Grouping Mean N gas A E B D B B C B B B A

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Lecture 9-1 Analysis of Variance

Lecture 9-1 Analysis of Variance

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