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Experimental StatisticsExperimental Statistics - week 4 - week 4Experimental StatisticsExperimental Statistics - week 4 - week 4

Chapter 8: 1-factor ANOVA models

Using SAS

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EXAM SCHEDULE: 

Exam I – Take-home exam (handed out Thursday, March 3, due 8:00 AM Tuesday, March 8) 

Exam II – Take-home exam (handed out Thursday, April 14, due 8:00 AM Tuesday, April 19) 

Final Exam – optional (scheduled for 8:00 AM – 11:00 AM Friday, May 6)  

GRADE COMPUTATION: 

Exam Grades (75%)Daily Assignments (25%)

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ANOVA Table Output - hostility data - calculations done in class 

 

Source SS df MS F p-value 

Between 767.17 2 383.58 16.7 <.001  samples

Within 205.74 9 22.86  samples

Totals 972.91 

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SPSS ANOVA Table for Hostility Data

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ANOVA Models

Consider the random sample

Population has mean .

1 2, ,..., ny y y

1 2 35.5, 3.8, 6.0,y y y where etc.

1 2, ,...,

,

, 1,...,

n

i i

y y y

y i n

2

If is a sample from a population that is

normal with mean and variance then we

can write

Note:

Example:

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11 1 11

12 1 12

y

y

We can write

etc.

For 1-factor ANOVA

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Alternative form of the 1-Factor ANOVA Model

2 ' are (0, )ij s NID

General Form of Model: ij i ijy

(pages 394-395)

- random errors follow a Normal (N) distribution, are independently distributed (ID), and have zero mean and constant variance

1

0t

ii

Note:

i i

ij i ijy

1

1

t

iit

-- i.e. variability does not change from group to group

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0 1 2:

:t

a

H

H

Testing the hypotheses:

at least 2 means a unequal

0 :

:a

H

H

is equivalent to testing the hypotheses:

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Analysis of Variance TableAnalysis of Variance TableAnalysis of Variance TableAnalysis of Variance Table

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0 2( 1, )B

TW

sH F F t n t

s We reject at significance level if

1F - if factor effects, we expect

2B is 22 estimates constant -

1F - if no factor effects, we expect ;

Recall:

In our model:2 2Ws estimates

Introduction to SAS Introduction to SAS Programming LanguageProgramming Language

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Recall CAR DATA

For this analysis, 5 gasoline types (A - E) were to be tested. Twenty carswere selected for testing and were assigned randomly to the groups (i.e. the gasoline types). Thus, in the analysis, each gasoline type was tested on 4 cars. A performance-based octane reading was obtained for each car, and the question is whether the gasolines differ with respect to this octane reading.  

  A

91.7 91.2 90.9 90.6

B

91.7 91.9 90.9 90.9

C

92.4 91.2 91.6 91.0

D

91.8 92.2 92.0 91.4

E

93.1 92.9 92.4 92.4

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 The CAR data set as SAS needs to see it:  A 91.7A 91.2A 90.9A 90.6B 91.7B 91.9B 90.9B 90.9C 92.4C 91.2C 91.6C 91.0D 91.8D 92.2D 92.0D 91.4E 93.1E 92.9E 92.4E 92.4

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Case 1:  Data within SAS FILE : DATA one;INPUT gas$ octane;DATALINES;A 91.7A 91.2 . . . E 92.4E 92.4 ;PROC GLM; (or ANOVA) CLASS gas; MODEL octane=gas; TITLE 'Gasoline Example - Completely Randomized Design'; MEANS gas/duncans;RUN;PROC MEANS mean var;RUN;PROC MEANS mean var;class gas;RUN;

SAS file for CAR data

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Brief Discussion of Components of the SAS File:

DATA Step

  DATA STATEMENT - the first DATA statement names the data set whose variables are defined in the INPUT statement -- in the above, we create data set 'one'

   INPUT STATEMENT - 2 forms

1.  Freefield - can be used when data values are separated by 1 or more blanks

       INPUT   NAME $  AGE SEX $   SCORE;          ($ indicates character variable)

  2.  Formatted - data occur in fixed columns

       INPUT    NAME $ 1-20  AGE 22-24  SEX  $ 26   SCORE 28-30;  

DATALINES STATEMENT       -  used to indicate that the next records in the file contain the actual data and the semicolon after the data indicates the end of the data itself  

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SPECIFYING THE ANALYSISSPECIFYING THE ANALYSIS --  PROC STATEMENTS

 GENERAL FORM   PROC xxxxx; implies procedure is to be run on most recently created data set  PROC xxxxx  DATA = data set name; Note:  I did not have to specify DATA=one in the above example

  Example PROCs:

PROC REG - regression analysisPROC ANOVA - analysis of variance PROC GLM - general linear model PROC MEANS - basic statistics, t-test for H0:

PROC PLOT - plottingPROC TTEST - t-tests PROC UNIVARIATE - descriptive stats, box-plots, etc.

PROC BOXPLOT - boxplots

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PROC GLMPROC GLMPROC GLMPROC GLM

• Proc GLM data = fn ;

– Class … ; List all the factors.

– Model … / options; e.g., model octane = gas;

– Means … / options;

– Run;

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SAS SyntaxSAS SyntaxSAS SyntaxSAS Syntax

• Every command MUSTMUST end with a semicolon– Commands can continue over two or more lines

• Variable names are 1-8 characters (letters and numerals, beginning with a letter or underscore), but no blanks or special characters

– Note: values for character variables can exceed 8 characters

• Comments – Begin with *, end with ;

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Titles and LabelsTitles and LabelsTitles and LabelsTitles and Labels

• TITLE ‘…’ ;– Up to 10 title lines: TITLE ‘include your title here’;

– Can be placed in Data Steps or Procs

• LABEL name = ‘…’ ;– Can be in a DATA STEP or PROC PRINT

– Include ALL labels, then a single ;

Note: For class assignments, place descriptive titles and labels on the output. Print the data to the output file.

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Case 2:  Data in an External File

FILENAME f1 ‘complete directory/file specification’;  

FILENAME f1 ‘a:car.data';DATA one;INFILE f1; INPUT gas$ octane;PROC GLM; (or ANOVA) CLASS gas; MODEL octane=gas; TITLE 'Gasoline Example - Completely Randomized Design';RUN;PROC MEANS mean var;RUN;PROC MEANS mean var;class gas;run;

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The SAS Output for CAR data:   Gasoline Example - Completely Randomized Design   General Linear Models Procedure Dependent Variable: OCTANE Sum of MeanSource DF Squares Square F Value Pr > F Model 4 6.10800000 1.52700000 6.80 0.0025 Error 15 3.37000000 0.22466667 Corrected Total 19 9.47800000  R-Square C.V. Root MSE OCTANE Mean  0.644440 0.516836 0.4739902 91.710000  Source DF Type I SS Mean Square F Value Pr > F GAS 4 6.10800000 1.52700000 6.80 0.0025 Source DF Type III SS Mean Square F Value Pr > F GAS 4 6.10800000 1.52700000 6.80 0.0025 

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Text Format for ANOVA Table Output - car data 

 

Source SS df MS F p-value 

Between 6.108 4 1.527 6.80 0.0025  samples

Within 3.370 15 0.225  samples

Totals 9.478 19 

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PC SAS on Campus

Library

BIC

Student Center

http://support.sas.com/rnd/le/index.html

SAS Learning Edition $125

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1. Calculate the average, standard deviation, minimum, and maximum for the 20 octane readings. CS pp. 25 - 32

2. Graph a histogram of OCTANE. CS pp. 37

3. Calculate descriptive statistics in (1) above for OCTANE for each of the 5 gasolines. CS pp. 32-34

0 : A BH Run 4. t-test to test using GA S typesA and B. CS pp. 138-141

“Lab” AssignmentUsing CAR Data, run the following in this order with one set of code:

5. Plot side-by-side box plots for OCTANE for the 5 levels of the variable GAS

6. Compute a 1-factor ANOVA for the CAR data using only the first 3 GAS types. CS pp.150-155