IES 612/STA 4-573/STA 4-576

Spring 2005

 

Week 6 – IES612-week06-lecture.doc

 

ANOVA MODELS – models for comparing means of different treatments or populations

 

Recall your old friend the two-group pooled variance t-test

 

H0: m1 = m2 [two populations do NOT differ in mean response]

 

Ha: m1 ≠ m2

 

Assumptions/Data?

 

Data from population 1:   (y11, y12, . . . , y1n1)

 

Data from population 2:   (y21, y22, . . . , y2n2)

 

Assume Yij ~ independent N(mi, s2)

i = 1,2

j = 1, 2, …, ni

 

Another way of writing this is Yij = mi + eij with eij ~ independent N(0, s2)

 

In other words, the response of the “jth” observation in the “ith” population can be written in terms of the mean of the ith population + how this observation differs from the mean.  Does this look familiar?

 

Test Statistic?

  where 

 

The pooled variance looks like something from regression.  What?

 

How about your new friend, regression?

 

Can we test H0: m1 = m2 [two populations differ in mean response]?  Vs. Ha: m1 ≠ m2

 

Assumptions/Data?

 

Data from population 1:   (y11, y12, . . . , y1n1)

 

Data from population 2:   (y21, y22, . . . , y2n2)

 

Assume Yij ~ independent N(mi, s2)  i = 1,2,  j = 1, 2, …, ni

 

Another way of writing this is Yij = mi + eij with eij ~ independent N(0, s2)

 

Let X = 1 (if group 2) and X=0 (if group 1) and Y = b0 + b1 X + e

 

Then

Group 1:   Y = b0 + e

Group 2:   Y = b0 + b1 + e

 

Implying m1 =  b0 and m2 = b0 + b1 so  b1 = m2 - m1.  Thus, H0: m1 = m2 AND H0: b1 = 0 test the same hypothesis.

 

Example:  Comparing Two-group T-test, Regression test and one-way ANOVA test

 

options ls=80 formdlim=”-“ nocenter nodate;

data meat;

  input condition $ logcount @@;

  ivac = (condition=”vacuum”);

  imix = (condition=”mixed”);

  datalines;

vacuum 5.26  vacuum 5.44  vacuum 5.80

 mixed 7.41   mixed 7.33   mixed 7.04

;

 

ods html;

 

title “Log(bacteria count) for different packaging conditions”;

 

proc boxplot;

title2 “Boxplots of log(count)”;

  plot logcount*condition;

  run;

 

proc ttest;

title2 “T-test comparing mix to vacuum conditions”;

  class condition;

  var logcount;

  run;

 

proc reg;

title2 “Regression with indicator variable for mix condition”;

  model logcount = imix;

  run;

 

proc glm;

title2 “One-way anova model”;

  class condition;

  model logcount = condition;

run;

 

ods html close;

 

Page 1 of PLOT for logcount.

 

 

Log(bacteria count) for different packaging conditions

T-test comparing mix to vacuum conditions

 

The TTEST Procedure

Statistics

Variable

condition

N

Lower CL
Mean

Mean

Upper CL
Mean

Lower CL
Std Dev

Std Dev

Upper CL
Std Dev

Std Err

Minimum

Maximum

logcount

mixed

3

6.7764

7.26

7.7436

0.1014

0.1947

1.2235

0.1124

7.04

7.41

logcount

vacuum

3

4.817

5.5

6.183

0.1432

0.275

1.728

0.1587

5.26

5.8

logcount

Diff (1-2)

 

1.22

1.76

2.3

0.1427

0.2382

0.6845

0.1945

 

 

 

T-Tests

Variable

Method

Variances

DF

t Value

Pr > |t|

logcount

Pooled

Equal

4

9.05

0.0008

logcount

Satterthwaite

Unequal

3.6

9.05

0.0013

 

Equality of Variances

Variable

Method

Num DF

Den DF

F Value

Pr > F

logcount

Folded F

2

2

1.99

0.6678

 

 

Log(bacteria count) for different packaging conditions

Regression with indicator variable for mix condition

 

The REG Procedure

Model: MODEL1

Dependent Variable: logcount

Number of Observations Read

6

Number of Observations Used

6

 

Analysis of Variance

Source

DF

Sum of
Squares

Mean
Square

F Value

Pr > F

Model

1

4.64640

4.64640

81.87

0.0008

Error

4

0.22700

0.05675

 

 

Corrected Total

5

4.87340

 

 

 

 

Root MSE

0.23822

R-Square

0.9534

Dependent Mean

6.38000

Adj R-Sq

0.9418

Coeff Var

3.73390

 

 

 

Parameter Estimates

Variable

DF

Parameter
Estimate

Standard
Error

t Value

Pr > |t|

Intercept

1

5.50000

0.13754

39.99

<.0001

imix

1

1.76000

0.19451

9.05

0.0008

 

 

Log(bacteria count) for different packaging conditions

One-way anova model

 

The GLM Procedure

Class Level Information

Class

Levels

Values

condition

2

mixed vacuum

 

Number of Observations Read

6

Number of Observations Used

6

 

Dependent Variable: logcount

Source

DF

Sum of Squares

Mean Square

F Value

Pr > F

Model

1

4.64640000

4.64640000

81.87

0.0008

Error

4

0.22700000

0.05675000

 

 

Corrected Total

5

4.87340000

 

 

 

 

R-Square

Coeff Var

Root MSE

logcount Mean

0.953421

3.733896

0.238223

6.380000

 

Source

DF

Type I SS

Mean Square

F Value

Pr > F

condition

1

4.64640000

4.64640000

81.87

0.0008

 

Source

DF

Type III SS

Mean Square

F Value

Pr > F

condition

1

4.64640000

4.64640000

81.87

0.0008

 

 

 

Test statistic

P-value

Comment

T-test

tobs=9.05

 

0.0008

Test of  “m1 = m2“ – note unequal variance t-test has same value test statistic [b/c sample sizes are the same]; however, slight modification in degrees of freedom

Regression

tobs=9.05

Fobs=81.87

 

0.0008

Test of “b1=0” in model

logcount =b0 + b1 I[condition=”mix”] + e

One-way ANOVA

Fobs=81.87

 

0.0008

Test of  “m1 = m2“

 

 

 

 

A more general formulation …

 

Numeric data – samples from “t’ populations obtained

 

(y11, y12, . . . , y1n1) = {y1j} j=1, …, n1

(y21, y22, . . . , y2n2) = {y2j} j=1, …, n2

…

(yt1, yt2, . . . , ytnt) = {ytj} j=1, …, nt

 

Assume Yij ~ independent N(mi, s2)

 

ni = number of observations from the ith population

 

i  = 1,2, …, t (populations or treatments)

j = 1, 2, …, ni (observations)

 

 

Terminology

 

* Designed experiments versus observational studies

* Completely Randomized Designs (CRD)

 

H0: m1 = m2= m3= … = mt

 

Ha: mi ≠ mj [at least two population means differ]

 

Assumptions/Data?

 

Assume Yij ~ independent N(mi, s2)

i = 1,2, …, t

j = 1, 2, …, ni

 

Test Statistic?

 

where the between(among) group variability is  and the within group variability is  

 

Reject H0 if

 

AOV Table

 

Source

SS

df

MS

Fobs

Between

SSB

t-1

SSB/(t-1)

MSB/MSW

Within

SSW

nT-t

SSW/( nT-t)

 

Totals

TSS

nT-1

 

 

 

 

Example  Bacteria growth in meat under different packaging conditions (revisited)

 

*--------------------------------------------------------------------;

title “One-way ANOVA/ CRD example + contrasts + multiple comparisons”;

title2 “Bacteria in meat data”;

data meat;

  input condition $ logcount @@;

  ivac = (condition=”vacuum”);

  imix = (condition=”mixed”);

  iCO2 = (condition=”CO2”);

  cards;

plastic 7.66 plastic 6.98 plastic 7.80

 vacuum 5.26  vacuum 5.44  vacuum 5.80

  mixed 7.41   mixed 7.33   mixed 7.04

    CO2 3.51     CO2 2.91     CO2 3.66

;

 

proc print data=meat;

  run;

 

proc sort out=smeat; by condition;

proc univariate plot; by condition;

title3 summary statistics and boxplot;

  var logcount;

  run;

 

proc reg data=meat;

title3  Regression with indicators;

  model logcount = ivac imix iCO2;

  run;

 

proc glm data=meat order=data;

title3  One-way anova + contrast + model adequacy;

  class condition;

  model logcount=condition;

  output out=new p=yhat r=resid;

  contrast 'plastic vs. rest' condition 3 -1 -1 -1;

  estimate 'plastic vs. rest' condition 3 -1 -1 -1;

  contrast 'CO2 vs. plastic'  condition -1  0  0 1;

  estimate 'CO2 vs. plastic'  condition -1  0  0 1;

  contrast 'CO2 vs. vacuum'   condition  0 -1  0 1;

  estimate 'CO2 vs. vacuum'   condition  0 -1  0 1;

  contrast 'CO2 vs. mixed'    condition  0  0 -1 1;

  estimate 'CO2 vs. mixed'    condition  0  0 -1 1;

  lsmeans condition / stderr pdiff;

  means condition / lsd clm;

  means condition / bon scheffe tukey;

  means condition / bon tukey cldiff;

  run;

 

proc plot data=new;

  plot logcount*condition yhat*condition='p' /overlay;

  plot resid*condition resid*yhat / vref=0;

  run;

 

proc univariate plot;

  var resid;

  run;

 

* construct the normal scores - Z[(i-.375)/(n+.25)];

* note not multiplied by sqrt(mse);

proc rank data=new normal=blom out=rnew;

  var resid;

  ranks nscore;

 

* generate plot analogous to univariate's normal prob. plot;

 

proc plot;

  plot resid*nscore;

  run;

 

data moremeat; set meat;

  count = exp(logcount);

  title3 raw count data analyzed;

 

proc glm data=moremeat;

  class condition;

  model count=condition;

  output out=mnew p=yhat r=resid;

  lsmeans condition / stderr pdiff;

*  means condition / clm bon scheffe lsd tukey snk;

 

proc plot data=mnew;

  plot count*condition yhat*condition='p' /overlay;

  plot resid*condition resid*yhat / vref=0;

 

proc univariate data=mnew plot;

  var resid;

 

proc rank data=mnew normal=blom out=rnew;

  var resid;

  ranks nscore;

proc plot;

  plot resid*nscore;

proc print data=meat;

  run;

 

Obs    condition    logcount    ivac    imix    iCO2

  1    plastic      7.66        0       0       0

  2    plastic      6.98        0       0       0

  3    plastic      7.80        0       0       0

  4    vacuum       5.26        1       0       0

  5    vacuum       5.44        1       0       0

  6    vacuum       5.80        1       0       0

  7    mixed        7.41        0       1       0

  8    mixed        7.33        0       1       0

  9    mixed        7.04        0       1       0

 10    CO2          3.51        0       0       1

 11    CO2          2.91        0       0       1

 12    CO2          3.66        0       0       1

 

proc sort out=smeat; by condition;

proc univariate plot; by condition;

title3 summary statistics and boxplot;

  var logcount;

  run;

The UNIVARIATE Procedure

Variable:  logcount

 

Schematic Plots

       8 +

         |                                 *-----*

         |                     +-----+     |  +  |

         |                     *--+--*     |     |

       7 +                     +-----+     +-----+

         |

         |

         |

       6 +

         |                                             +-----+

         |                                             *--+--*

         |                                             +-----+

       5 +

         |

         |

         |

       4 +

         |         +-----+

         |         *-----*

         |         |  +  |

       3 +         +-----+

         |

         |

         |

       2 +

          ------------+-----------+-----------+-----------+-----------

 condition               CO2       mixed     plastic      vacuum

 

proc reg data=meat;

title3  Regression with indicators;

  model logcount = ivac imix iCO2;

  run;

 

The REG Procedure

Model: MODEL1

Dependent Variable: logcount

 

                             Analysis of Variance

 

                                    Sum of           Mean

Source                   DF        Squares         Square    F Value    Pr > F

Model                     3       32.87280       10.95760      94.58    <.0001

Error                     8        0.92680        0.11585

Corrected Total          11       33.79960

 

 

Root MSE              0.34037    R-Square     0.9726

Dependent Mean        5.90000    Adj R-Sq     0.9623

Coeff Var             5.76894

 

                        Parameter Estimates

 

                     Parameter       Standard

Variable     DF       Estimate          Error    t Value    Pr > |t|

 

Intercept     1        7.48000        0.19651      38.06      <.0001

ivac          1       -1.98000        0.27791      -7.12      <.0001

imix          1       -0.22000        0.27791      -0.79      0.4514

iCO2          1       -4.12000        0.27791     -14.83      <.0001

 

proc glm data=meat order=data;

title3  One-way anova + contrast + model adequacy;

  class condition;

  model logcount=condition;

  output out=new p=yhat r=resid;

 

The GLM Procedure

             Class Level Information

Class         Levels    Values

condition            4    plastic vacuum mixed CO2

 

Number of observations    1

The GLM Procedure

Dependent Variable: logcount

                                        Sum of

Source                      DF         Squares     Mean Square    F Value    Pr > F

Model                        3     32.87280000     10.95760000      94.58    <.0001

Error                        8      0.92680000      0.11585000

Corrected Total             11     33.79960000

 

R-Square     Coeff Var      Root MSE    logcount Mean

0.972580      5.768940      0.340367         5.900000

 

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

condition                      3     32.87280000     10.95760000      94.58    <.0001

 

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

condition                      3     32.87280000     10.95760000      94.58    <.0001

 

Contrast                    DF     Contrast SS     Mean Square    F Value    Pr > F

plastic vs. rest             1      9.98560000      9.98560000      86.19    <.0001

CO2 vs. plastic              1     25.46160000     25.46160000     219.78    <.0001

CO2 vs. vacuum               1      6.86940000      6.86940000      59.30    <.0001

CO2 vs. mixed                1     22.81500000     22.81500000     196.94    <.0001

 

  contrast 'plastic vs. rest' condition 3 -1 -1 -1;

  estimate 'plastic vs. rest' condition 3 -1 -1 -1;

  contrast 'CO2 vs. plastic'  condition -1  0  0 1;

  estimate 'CO2 vs. plastic'  condition -1  0  0 1;

  contrast 'CO2 vs. vacuum'   condition  0 -1  0 1;

  estimate 'CO2 vs. vacuum'   condition  0 -1  0 1;

  contrast 'CO2 vs. mixed'    condition  0  0 -1 1;

  estimate 'CO2 vs. mixed'    condition  0  0 -1 1;

Dependent Variable: logcount

 

                                            Standard

Parameter                   Estimate           Error    t Value    Pr > |t|

 

plastic vs. rest          6.32000000      0.68073490       9.28      <.0001

CO2 vs. plastic          -4.12000000      0.27790886     -14.83      <.0001

CO2 vs. vacuum           -2.14000000      0.27790886      -7.70      <.0001

CO2 vs. mixed            -3.90000000      0.27790886     -14.03      <.0001

 

  lsmeans condition / stderr pdiff;

 

The GLM Procedure

Least Squares Means

 

               logcount        Standard                  LSMEAN

condition          LSMEAN           Error    Pr > |t|      Number

plastic      7.48000000      0.19651124      <.0001           1

vacuum       5.50000000      0.19651124      <.0001           2

mixed        7.26000000      0.19651124      <.0001           3

CO2          3.36000000      0.19651124      <.0001           4

 

           Least Squares Means for effect condition

            Pr > |t| for H0: LSMean(i)=LSMean(j)

                Dependent Variable: logcount

i/j              1             2             3             4

   1                      <.0001        0.4514        <.0001

   2        <.0001                      0.0002        <.0001

   3        0.4514        0.0002                      <.0001

   4        <.0001        <.0001        <.0001

 

NOTE: To ensure overall protection level, only probabilities associated with pre-planned comparisons should be used

  means condition / lsd clm;

 

t Confidence Intervals for logcount

 

Alpha                                 0.05

Error Degrees of Freedom                 8

Error Mean Square                  0.11585

Critical Value of t                2.30600

Half Width of Confidence Interval 0.453156

 

                                      95% Confidence

condition         N          Mean           Limits

plastic         3        7.4800      7.0268      7.9332

mixed           3        7.2600      6.8068      7.7132

vacuum          3        5.5000      5.0468      5.9532

CO2             3        3.3600      2.9068      3.8132

 

  means condition / bon scheffe tukey;

Tukey's Studentized Range (HSD) Test for logcount

 

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                   8

Error Mean Square                    0.11585

Critical Value of Studentized Range  4.52880

Minimum Significant Difference          0.89

 

Means with the same letter are not significantly different.

 

           Mean      N    condition

A        7.4800      3    plastic

A

A        7.2600      3    mixed

 

B        5.5000      3    vacuum

 

C        3.3600      3    CO2

 

Bonferroni (Dunn) t Tests for logcount

 

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              8

Error Mean Square               0.11585

Critical Value of t             3.47888

Minimum Significant Difference   0.9668

 

Means with the same letter are not significantly different.

 

           Mean      N    condition

 

A        7.4800      3    plastic

A

A        7.2600      3    mixed

 

B        5.5000      3    vacuum

 

C        3.3600      3    CO2

 

Scheffe's Test for logcount

NOTE: This test controls the Type I experimentwise error rate.

 

Alpha                              0.05

Error Degrees of Freedom              8

Error Mean Square               0.11585

Critical Value of F             4.06618

Minimum Significant Difference   0.9706

 

Means with the same letter are not significantly different.

           Mean      N    condition

A        7.4800      3    plastic

A

A        7.2600      3    mixed

 

B        5.5000      3    vacuum

 

C        3.3600      3    CO

 

  means condition / bon tukey cldiff;

 

Tukey's Studentized Range (HSD) Test for logcount

NOTE: This test controls the Type I experimentwise error rate.

Alpha                                   0.05

Error Degrees of Freedom                   8

Error Mean Square                    0.11585

Critical Value of Studentized Range  4.52880

Minimum Significant Difference          0.89

 

Comparisons significant at the 0.05 level are indicated by ***.

 

                     Difference

     condition            Between     Simultaneous 95%

   Comparison             Means    Confidence Limits

plastic - mixed          0.2200     -0.6700   1.1100

plastic - vacuum         1.9800      1.0900   2.8700  ***

plastic - CO2            4.1200      3.2300   5.0100  ***

mixed   - plastic       -0.2200     -1.1100   0.6700

mixed   - vacuum         1.7600      0.8700   2.6500  ***

mixed   - CO2            3.9000      3.0100   4.7900  ***

vacuum  - plastic       -1.9800     -2.8700  -1.0900  ***

vacuum  - mixed         -1.7600     -2.6500  -0.8700  ***

vacuum  - CO2            2.1400      1.2500   3.0300  ***

CO2     - plastic       -4.1200     -5.0100  -3.2300  ***

CO2     - mixed         -3.9000     -4.7900  -3.0100  ***

CO2     - vacuum        -2.1400     -3.0300  -1.2500  **

 

Bonferroni (Dunn) t Tests for logcount

 

NOTE: This test controls the Type I experimentwise error rate, but it generally has a higher Type II error rate than Tukey's for all pairwise comparisons.

Alpha                              0.05

Error Degrees of Freedom              8

Error Mean Square               0.11585

Critical Value of t             3.47888

Minimum Significant Difference   0.9668

 

Comparisons significant at the 0.05 level are indicated by ***.

                     Difference

     condition            Between     Simultaneous 95%

   Comparison             Means    Confidence Limits

plastic - mixed          0.2200     -0.7468   1.1868

plastic - vacuum         1.9800      1.0132   2.9468  ***

plastic - CO2            4.1200      3.1532   5.0868  ***

mixed   - plastic       -0.2200     -1.1868   0.7468

mixed   - vacuum         1.7600      0.7932   2.7268  ***

mixed   - CO2            3.9000      2.9332   4.8668  ***

vacuum  - plastic       -1.9800     -2.9468  -1.0132  ***

vacuum  - mixed         -1.7600     -2.7268  -0.7932  ***

vacuum  - CO2            2.1400      1.1732   3.1068  ***

CO2     - plastic       -4.1200     -5.0868  -3.1532  ***

CO2     - mixed         -3.9000     -4.8668  -2.9332  ***

CO2     - vacuum        -2.1400     -3.1068  -1.1732  ***

 

options ls=70;

proc plot data=new;

  plot logcount*condition yhat*condition='p' /overlay;

  plot resid*condition resid*yhat / vref=0;

run;

      Plot of resid*condition.  Legend: A = 1 obs, B = 2 obs, etc.

     resid ‚

           ‚

       0.4 ˆ

           ‚

           ‚  A                               A               A

           ‚

           ‚

       0.2 ˆ                                  A

           ‚  A               A

           ‚

           ‚                  A

           ‚

       0.0 ˆƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ

           ‚                                                  A

           ‚

           ‚

           ‚

      -0.2 ˆ                  A

           ‚                                                  A

           ‚

           ‚

           ‚

      -0.4 ˆ

           ‚  A

           ‚                                  A

           ‚

           ‚

      -0.6 ˆ

           ‚

           Šƒƒˆƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒˆƒƒ

             CO2            mixed          plastic         vacuum

 

                                   Condition

       Plot of resid*yhat.  Legend: A = 1 obs, B = 2 obs, etc.

 

resid ‚

      ‚

  0.4 ˆ

      ‚

      ‚     A                         A                       A

      ‚

      ‚

  0.2 ˆ                                                       A

      ‚     A                                              A

      ‚

      ‚                                                    A

      ‚

  0.0 ˆƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ

      ‚                               A

      ‚

      ‚

      ‚

 -0.2 ˆ                                                    A

      ‚                               A

      ‚

      ‚

      ‚

 -0.4 ˆ

      ‚     A

      ‚                                                       A

      ‚

      ‚

 -0.6 ˆ

      ‚

      Šƒˆƒƒƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒƒƒˆƒ

        3           4           5           6           7           8

 

                                    yhat

 

proc univariate plot;

  var resid;

 

* construct the normal scores - Z[(i-.375)/(n+.25)];

* note not multiplied by sqrt(mse);

proc rank data=new normal=blom out=rnew;

  var resid;

  ranks nscore;

 

* generate plot analogous to univariate's normal prob. plot;

 

proc plot;

  plot resid*nscore;

      Plot of resid*nscore.  Legend: A = 1 obs, B = 2 obs, etc.

 

     resid ‚

           ‚

       0.4 ˆ

           ‚

           ‚                                     B        A

           ‚

           ‚

       0.2 ˆ                                A

           ‚                           A  A

           ‚

           ‚                         A

           ‚

       0.0 ˆ

           ‚                      A

           ‚

           ‚

           ‚

      -0.2 ˆ                    A

           ‚                 A

           ‚

           ‚

           ‚

      -0.4 ˆ

           ‚             A

           ‚      A

           ‚

           ‚

      -0.6 ˆ

           ‚

           Šƒƒˆƒƒƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒƒƒˆƒƒ

             -2          -1           0           1           2

 

                           Rank for Variable resid

 

data moremeat; set meat;

  count = exp(logcount);

  title3 raw count data analyzed;

 

proc glm data=moremeat;

  class condition;

  model count=condition;

  output out=mnew p=yhat r=resid;

  lsmeans condition / stderr pdiff;

*  means condition / clm bon scheffe lsd tukey snk;

  run;

 

The GLM Procedure

Dependent Variable: count

                                        Sum of

Source                      DF         Squares     Mean Square    F Value    Pr > F

Model                        3     7282652.348     2427550.783      16.56    0.0009

Error                        8     1172820.616      146602.577

Corrected Total             11     8455472.964

 

R-Square     Coeff Var      Root MSE    count Mean

0.861294      42.54159      382.8872      900.0303

 

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

condition                      3     7282652.348     2427550.783      16.56    0.0009

 

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

condition                      3     7282652.348     2427550.783      16.56    0.0009

 

proc plot data=mnew;

  plot count*condition yhat*condition='p' /overlay;

  plot resid*condition resid*yhat / vref=0;

  run;

 

    Plot of resid*condition.  Legend: A = 1 obs, B = 2 obs, etc.

 

     resid ‚

      1000 ˆ

           ‚

           ‚

           ‚

           ‚

           ‚

           ‚                                  A

       500 ˆ

           ‚

           ‚

           ‚

           ‚                  A               A

           ‚

           ‚                  A                               A

         0 ˆƒƒCƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒAƒƒ

           ‚                                                  A

           ‚

           ‚

           ‚                  A

           ‚

           ‚

      -500 ˆ

           ‚

           ‚

           ‚

           ‚                                  A

           ‚

           ‚

     -1000 ˆ

           Šƒƒˆƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒˆƒƒ

             CO2            mixed          plastic         vacuum

 

                                   Condition

       Plot of resid*yhat.  Legend: A = 1 obs, B = 2 obs, etc.

 

     resid ‚

      1000 ˆ

           ‚

           ‚

           ‚

           ‚

           ‚

           ‚                                               A

       500 ˆ

           ‚

           ‚

           ‚

           ‚                                     A         A

           ‚

           ‚        A                            A

         0 ˆƒƒƒCƒƒƒƒAƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ

           ‚        A

           ‚

           ‚

           ‚                                     A

           ‚

           ‚

      -500 ˆ

           ‚

           ‚

           ‚

           ‚                                               A

           ‚

           ‚

     -1000 ˆ

           Šƒƒˆƒƒƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒƒƒˆƒƒ

              0          500        1000        1500        2000

 

                                    yhat

 

proc rank data=mnew normal=blom out=rnew;

  var resid;

  ranks nscore;

proc plot;

  plot resid*nscore;

  run;

 

      Plot of resid*nscore.  Legend: A = 1 obs, B = 2 obs, etc.

 

     resid ‚

      1000 ˆ

           ‚

           ‚

           ‚

           ‚

           ‚

           ‚                                              A

       500 ˆ

           ‚

           ‚

           ‚

           ‚                                   A   A

           ‚

           ‚                              A A

         0 ˆ                    A A  A A

           ‚                 A

           ‚

           ‚

           ‚             A

           ‚

           ‚

      -500 ˆ

           ‚

           ‚

           ‚

           ‚      A

           ‚

           ‚

     -1000 ˆ

           Šƒƒˆƒƒƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒƒƒˆƒƒ

             -2          -1           0           1           2

 

                           Rank for Variable resid