5-5 Inference on the Ratio of Variances of Two Normal Populations

5-5.1 The F DistributionWe wish to test the hypotheses:

• The development of a test procedure for these hypotheses requires a new probability distribution, the F distribution.

5-5.1 The F Distribution

The Test Procedure

Example 5-10 OPTIONS NOOVP NODATE NONUMBER LS=80;DATA ex510;n1=16; n2=16; alpha=0.05;df1=n1-1; df2=n2-1;s1=1.96; s2=2.13;f0=(s1**2)/(s2**2);If f0 <= 1 then pvalue=2*probf(f0, df1, df2); /*f0=0.85 less than 1.0 */

Else pvalue=2*(1-probf(f0, df1, df2));f1=finv(alpha/2, df1, df2); f2=1/finv(alpha/2, df2, df1);fl=finv(alpha/2, df2, df1);fu=finv(1-alpha/2, df2, df1);CL=f0*fl;CU=f0*fu;PROC PRINT;var f0 df1 df2 pvalue f1 f2 fl fu CL CU;RUN; QUIT;

SAS 시스템

OBS f0 df1 df2 pvalue f1 f2 fl fu CL CU

1 0.84675 15 15 0.75151 0.34939 2.86209 0.34939 2.86209 0.29585 2.42346

5-5.2 Confidence Interval on the Ratio of Two Variances

5-6 Inference on Two Population Proportions

5-6.1 Hypothesis Testing on the Equality of Two Binomial Proportions

5-6 Inference on Two Population Proportions 5-6.1 Hypothesis Testing on the Equality of Two Binomial Proportions

OPTIONS NOOVP NODATE NONUMBER LS=80;DATA EX512;N1=300; N2=300; ALPHA=0.05;DFT1=253; DFT2=196;P1HAT=DFT1/N1; P2HAT=DFT2/N2;DIFFP=P1HAT-P2HAT;PHAT=(DFT1+DFT2)/(N1+N2);Z0=DIFFP/SQRT(PHAT*(1-pHAT)*(1/N1+1/N2));PVALUE=2*(1-PROBNORM(Z0));ZVALUE=ABS(PROBIT(ALPHA/2)); /*PROBNORM is the inverse of the PROBIT function */LIMIT=ZVALUE*SQRT((P1HAT*(1-P1HAT)/N1) + (P2HAT*(1-p2HAT)/N2));UL=DIFFP+LIMIT; LL=DIFFP-LIMIT;PROC PRINT;var P1HAT P2HAT DIFFP PHAT Z0 ZVALUE PVALUE LL UL;RUN; QUIT;

OBS P1HAT P2HAT DIFFP PHAT Z0 ZVALUE PVALUE LL UL 1 0.84333 0.65333 0.19 0.74833 5.36215 1.95996 8.2238E-8 0.12224 0.25776

5-6.2 Type II Error and Choice of Sample Size

5-8 What If We Have More Than Two Samples?

5-8.1 Completely Randomized Experiment and Analysis of Variance

(Replicates)

• The levels of the factor are sometimes called treatments.

• Each treatment has six observations or replicates.

• The runs are run in random order.

= + =0

Linear Statistical Model

Which means differ?

Multiple Comparisons

If we wish to make all pairwise comparisons we need to worry about the probability of a type I error on each test and also the probability of making at least one type I error on all the tests. This latter error is referred to as the experiment-wise error rate or the overall error rate. A number of procedures are available to protect the overall error rate. They all compare the differences to a cut-off value. The cut-off values all have the form

Steps in Multiple Comparisons

1) Calculate .2) Determine cut-off. Any difference in 1) larger than the cut-off have

corresponding means that are significantly different from one another.

A. Fisher’s Least Significant Difference (LSD). This does not protect the overall error rate. The overall error rate will be approximately , where c is the number of comparisons being made.

If all then

B. Tukey’s Highest Significance Difference (HSD). This controls overall error rate. The overall error rate will be a.

If all then An alternative in unbalanced designs is

With Tukey’s you need both the degrees of freedom (n – r) and also the number of means being compared (r).

C. Student’s, Neuman’s, Keuls (SNK). This overall error rate is between that of Fisher’s and

Tukey’s. d is the number of means apart the two means being

compared are in the ordered list of means (d between 2 and r). Notice, more than one cut-off is needed for the SNK. The SNK is done sequentially, starting with the means the farthest apart. Once a pair of means is found to be not significantly different, all pairs closer together are deemed not significantly different.

Residual Analysis and Model Checking

OPTIONS NOOVP NODATE NONUMBER LS=80;proc format; value hc 1=' 5%' 2='10%' 3='15%' 4='20%';DATA ex514;INPUT hc strength @@; format hc hc.;CARDS;1 7 2 12 3 14 4 19 1 8 2 17 3 18 4 25 1 15 2 13 3 19 4 22 1 11 2 18 3 17 4 231 9 2 19 3 16 4 18 1 10 2 15 3 18 4 20proc anova data=ex514; class hc; model strength= hc; means hc/lsd snk tukey;TITLE 'proc anova balanced 1-way anova';proc sort; by hc;proc boxplot; plot strength*hc;proc glm data=ex514; class hc; model strength= hc; means hc/lsd snk tukey; output out=new p=phat r=resid;TITLE 'proc glm 1-way anova';proc plot data=new; plot resid*(phat hc); Title 'Residual plot'; RUN; QUIT;

proc anova balanced 1-way anova The ANOVA Procedure Class Level Information Class Levels Values hc 4 5% 10% 15% 20% Number of Observations Read 24 Number of Observations Used 24

proc anova balanced 1-way anova

The ANOVA ProcedureDependent Variable: strength Sum of Source DF Squares Mean Square F Value Pr > F Model 3 382.7916667 127.5972222 19.61 <.0001 Error 20 130.1666667 6.5083333 Corrected Total 23 512.9583333

R-Square Coeff Var Root MSE strength Mean 0.746243 15.98628 2.551144 15.95833

Source DF Anova SS Mean Square F Value Pr > F hc 3 382.7916667 127.5972222 19.61 <.0001

The ANOVA Procedure t Tests (LSD) for strength

NOTE: This test controls the Type I comparisonwise error rate, not the experimentwise error rate.

Alpha 0.05 Error Degrees of Freedom 20 Error Mean Square 6.508333 Critical Value of t 2.08596 Least Significant Difference 3.0724

Means with the same letter are not significantly different.

t Grouping Mean N hc A 21.167 6 20% B 17.000 6 15%

B 15.667 6 10% C 10.000 6 5%

The ANOVA Procedure Student-Newman-Keuls Test for strength

NOTE: This test controls the Type I experimentwise error rate under the complete null hypothesis but not under partial null hypotheses.

Alpha 0.05 Error Degrees of Freedom 20 Error Mean Square 6.508333

Number of Means 2 3 4 Critical Range 3.0724227 3.726419 4.1225627

SNK Grouping Mean N hc A 21.167 6 20% B 17.000 6 15%

B 15.667 6 10% C 10.000 6 5%

The ANOVA Procedure Tukey's Studentized Range (HSD) Test for strength

NOTE: This test controls the Type I experimentwise error rate, but it generally has a higher Type II error rate than REGWQ.

Alpha 0.05 Error Degrees of Freedom 20 Error Mean Square 6.508333 Critical Value of Studentized Range 3.95829 Minimum Significant Difference 4.1226

Tukey Grouping Mean N hc A 21.167 6 20% B 17.000 6 15%

B 15.667 6 10% C 10.000 6 5%

proc glm 1-way anova

The GLM ProcedureDependent Variable: strength Sum of Source DF Squares Mean Square F Value Pr > F Model 3 382.7916667 127.5972222 19.61 <.0001 Error 20 130.1666667 6.5083333 Corrected Total 23 512.9583333

Source DF Type I SS Mean Square F Value Pr > F hc 3 382.7916667 127.5972222 19.61 <.0001

Source DF Type III SS Mean Square F Value Pr > F hc 3 382.7916667 127.5972222 19.61 <.0001

Residual plot

resid*phat 도표 . 범례 : A = 1 관측치 , B = 2 관측치 , 등 .

resid | | 5.000 + A 4.500 + 4.000 + 3.500 + A 3.000 + A 2.500 + A 2.000 + A 1.500 + A A 1.000 + A B 0.500 + A 0.000 + A A-0.500 + A-1.000 + A A-1.500 + A-2.000 + A-2.500 + A A-3.000 + A A-3.500 + A-3.667 + A | ---+----------+----------+----------+----------+----------+----------+-- 10 12 14 16 18 20 22

Residual plot

resid*hc 도표 . 범례 : A = 1 관측치 , B = 2 관측치 , 등 .

resid | | 5.000 + A 4.500 + 4.000 + A 3.500 + A 3.000 + 2.500 + A 2.000 + A 1.500 + A A 1.000 + A B 0.500 + A 0.000 + A A -0.500 + A -1.000 + A A -1.500 + A -2.000 + A A -2.500 + A -3.000 + A A -3.500 + A -3.667 + A | ---+-----------------+-----------------+-----------------+-- 5% 10% 15% 20%

Kruskal-Wallis Nonparametric Test

H0: All k populations have the same distributionH1: Not all populations have the same distribution.

Test Statistic:

Here the Ri are the sum of the ranks for each population. The ranks are found by ranking the data in all populations combined. The n is the total sample size and ni is the sample size of the sample from the ith population.

Rejection Region:Paired comparison of population i to j:

Example

An article in Fortune compared rent in five American cities: New York, Chicago, Detroit, Tampa, and Orlando. The following data are small random samples of rants (in dollars) in the five cities. The New York data are Manhattan only. Conduct the Kruskal-Wallis test to determine whether evidence exists that there are significant differences in the rents in these cities. If differences exit, where are they?

OPTIONS NOOVP NODATE NONUMBER LS=80;proc format; value city 1=' New York' 2='Chicago' 3='Detroit' 4='Tampa' 5='Orlando';DATA rent;INPUT city rent @@; format city city.;CARDS;1 900 1 1200 1 850 1 1320 1 1400 1 1150 1 9752 625 2 640 2 775 2 1000 2 690 2 550 2 840 2 7503 415 3 400 3 420 3 560 3 780 3 620 3 800 3 3904 410 4 310 4 320 4 280 4 500 4 385 4 4405 340 5 425 5 275 5 210 5 575 5 360ods graphics on;proc npar1way data=rent wilcoxon plots=wilcoxonboxplot; class city; var rent;TITLE 'Kruskal-Wallis Test'; run; ods graphics off;quit;

Kruskal-Wallis Test The NPAR1WAY Procedure Wilcoxon Scores (Rank Sums) for Variable rent Classified by Variable city

Sum of Expected Std Dev Mean city N Scores Under H0 Under H0 Score -------------------------------------------------------------------------------- New York 7 228.0 129.50 25.018327 32.571429 Chicago 8 192.0 148.00 26.280538 24.000000 Detroit 8 135.0 148.00 26.280538 16.875000 Tampa 7 62.0 129.50 25.018327 8.857143 Orlando 6 49.0 111.00 23.558438 8.166667

Kruskal-Wallis Test Chi-Square 26.4930 DF 4 Pr > Chi-Square <.0001

NYw 8 Chicago Detroit Tampa Orlando

228 192 135 62 49

49 179* 143* 86* 13 -

62 166* 130* 73* -

135 93* 57* -

192 36* -

Cutoff for pairwise comparisons = 18.06

* Significantly different.

5-8.2 Randomized Complete Block Experiment

The randomized block design is an extension of the paired t-test to situations where the factor of interest has more than two levels.

5-8.2 Randomized Complete Block ExperimentFor example, consider the situation where two different methods were used to predict the shear strength of steel plate girders. Say we use four girders as the experimental units.

The appropriate linear statistical model:

We assume

• treatments and blocks are initially fixed effects

• blocks do not interact

The hypotheses of interest are:

The mean squares are:

5-8.2 Randomized Complete Block ExperimentThe expected values of these mean squares are:

5-8.2 Randomized Complete Block Experiment The model for the randomized block design is the same as the two-

way factorial where the interaction terms are assumed to be zero. The main question of interest is whether the means are the same for the different treatment levels. The blocks are used to explain some of the variability and many times to simplify the mechanics of collecting the data. For instance if the blocks are cities, it is much easier to collect data in one city at a time. The SSTR is the same as SSA in the two-way factorial, SSBL is the same as the SSB in the two-way factorial. Notice SSE in the corresponding two-way factorial model has zero degrees of freedom when you have only one observation per cell. The two-way factorial with only one observation is analyzed the same as the randomized block, and also assumes no interaction between the two factors.

Which means differ?

5-8 What If We Have More Than Two Samples? Residual Analysis and Model Checking

OPTIONS NOOVP NODATE NONUMBER LS=80;DATA ex515;DO type=1 to 4; DO fabsam=1 TO 5; INPUT strength @@; OUTPUT; END; END;CARDS;1.3 1.6 0.5 1.2 1.1 2.2 2.4 0.4 2 1.8 1.8 1.7 0.6 1.5 1.3 3.9 4.4 2 4.1 3.4PROC GLM DATA=ex515; CLASS type fabsam; MODEL strength= type fabsam; MEANS TYPE/SNK; output out=new p=phat r=resid;TITLE 'Randomized Block Design FabSample: Block, ChemType: Treatment';proc sort; by type;proc boxplot; plot strength*type;proc plot data=new; plot resid*phat; /* Residual Plot */ plot resid*type; /* Residual by Chemical Type */ plot resid*fabsam; /* Residuals by block */proc anova data=ex515; class type; model strength=type; means type/snk;TITLE 'one-way anova';run; quit;

Example 5-15

Randomized Block Design FabSample: Block, ChemType: Treatment

The GLM Procedure

Class Level Information

Class Levels Values

type 4 1 2 3 4

fabsam 5 1 2 3 4 5

Number of Observations Read 20 Number of Observations Used 20

The GLM Procedure

Dependent Variable: strength

Sum of Source DF Squares Mean Square F Value Pr > F

Model 7 24.73700000 3.53385714 44.59 <.0001

Error 12 0.95100000 0.07925000

Corrected Total 19 25.68800000

R-Square Coeff Var Root MSE strength Mean

0.962979 14.36295 0.281514 1.960000

Source DF Type I SS Mean Square F Value Pr > F

type 3 18.04400000 6.01466667 75.89 <.0001 fabsam 4 6.69300000 1.67325000 21.11 <.0001

Source DF Type III SS Mean Square F Value Pr > F

type 3 18.04400000 6.01466667 75.89 <.0001 fabsam 4 6.69300000 1.67325000 21.11 <.0001

The ANOVA Procedure

Dependent Variable: strength

Sum of Source DF Squares Mean Square F Value Pr > F

Model 7 24.73700000 3.53385714 44.59 <.0001

Error 12 0.95100000 0.07925000

Corrected Total 19 25.68800000

R-Square Coeff Var Root MSE strength Mean

0.962979 14.36295 0.281514 1.960000

Source DF Anova SS Mean Square F Value Pr > F

type 3 18.04400000 6.01466667 75.89 <.0001 fabsam 4 6.69300000 1.67325000 21.11 <.0001

Randomized Block Design FabSample: Block, ChemType: Treatment The ANOVA Procedure Student-Newman-Keuls Test for strength NOTE: This test controls the Type I experimentwise error rate under the complete null hypothesis but not under partial null hypotheses.

SNK Grouping Mean N type A 3.5600 5 4 B 1.7600 5 2 C B 1.3800 5 3 C 1.1400 5 1

Randomized Block Design FabSample: Block, ChemType: Treatment The ANOVA Procedure Student-Newman-Keuls Test for strength NOTE: This test controls the Type I experimentwise error rate under the complete null hypothesis but not under partial null hypotheses.

SNK Grouping Mean N type A 3.5600 5 4 B 1.7600 5 2 C B 1.3800 5 3 C 1.1400 5 1

resid*phat 도표 . 범례 : A = 1 관측치 , B = 2 관측치 , 등 .resid | | 0.5 + | A | 0.4 + | | 0.3 + A A | A | 0.2 + | | 0.1 + A A | A A | A 0.0 + A A | A | -0.1 + A A | A | A A -0.2 + | A | A -0.3 + | | -0.4 + | | A -0.5 + | --+-------------+-------------+-------------+-------------+-------------+- 0 1 2 3 4 5 phat

Randomized Block Design FabSample: Block, ChemType: Treatment resid*fabsam 도표 . 범례 : A = 1 관측치 , B = 2 관측치 , 등 . resid | | 0.5 + | A | 0.4 + | | 0.3 + A A | A | 0.2 + | | 0.1 + A A | A A | A 0.0 + A A | A | -0.1 + A A | A | A A -0.2 + | A | A -0.3 + | | -0.4 + | | A -0.5 + | ---+--------------+--------------+--------------+--------------+-- 1 2 3 4 5

fabsam

Randomized Block Design FabSample: Block, ChemType: Treatment resid*type 도표 . 범례 : A = 1 관측치 , B = 2 관측치 , 등 . resid | | 0.5 + | A | 0.4 + | | 0.3 + A A | A | 0.2 + | | 0.1 + B | A A | A 0.0 + A A | A | -0.1 + A A | A | B -0.2 + | A | A -0.3 + | | -0.4 + | | A -0.5 + | ---+------------------+------------------+------------------+-- 1 2 3 4

one-way anova The ANOVA ProcedureDependent Variable: strength Sum of Source DF Squares Mean Square F Value Pr > F Model 3 18.04400000 6.01466667 12.59 0.0002 Error 16 7.64400000 0.47775000 Corrected Total 19 25.68800000

Source DF Anova SS Mean Square F Value Pr > F type 3 18.04400000 6.01466667 12.59 0.0002

one-way anova The ANOVA Procedure Student-Newman-Keuls Test for strength NOTE: This test controls the Type I experimentwise error rate under the complete null hypothesis but not under partial null hypotheses.

SNK Grouping Mean N type A 3.5600 5 4 B 1.7600 5 2 B 1.3800 5 3

B 1.1400 5 1

5-5 Inference on the Ratio of Variances of Two Normal Populations

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