SAS Library Slicing Interactions in SAS

This page was adapted from a page created by Oliver Schabenberger. We thank Professor Schabenberger for permission to adapt and distribute this page via our web site.

Introduction

In factorial experiments significant interactions are perceived as a real problem. Some analysts feel that the significance of an interaction is detrimental to interpretation of the results. These fears are unfounded. A problem in the past has been that little guidance and tools were available to proceed with post-ANOVA procedures in the presence of significant interactions. This has dramatically changed with recent implementations in The SAS System. The purpose of this page is to introduce the interested reader to the technique of slicing, the interpretation of effect slices and appropriate post-ANOVA procedures in light of significant interactions. Among statisticians the term slicing is not common. It is actually derived from SAS parlance. The correct terms for the procedure are simple effects or simple main effects.

If factors A and B interact one interpretation is as a lack of additivity. This stems from the linear model representation of the factorial treatment and design structure. If two factors act independently on the outcome of an experiment, their contributions are additive. A makes its contribution independent of B and you can add the two up to assess their joint effect. If an interaction between A and B is present this additivity no longer applies. You have to know the particular level of B to assess the effects of A and vice versa. An alternative interpretation is

Comparisons among the levels of A depend on the level of B they are performed at; Comparisons among the levels of B depend on the level of A they are performed at.

If, for example, A refers to different locations and B to a treatment factor, their interaction implies that there may be differences among the B levels (= a treatment effect) at some locations, while a treatment effect is absent at others. Consequently, one should look at the treatment comparisons separately by location.

Physical slicing

A physical slice implies breaking the data set into separate parts. If there are three locations (factor A) and four treatment levels (factor B) you separate the data into three different sets and run one-way ANOVAs for each location. In SAS this is done easily with by-processing, e.g.,

data exper1; input block location treat y; datalines; <data goes here> ; run; proc sort data=exper1; by location; run; proc glm data=exper1; class block treat; model y = block treat; by location; run;

What are the disadvantages of physical slicing?

1. The power of an ANOVA F-test is a function of the error degrees of freedom, i.e., the amount of information used to estimate the error mean square. By physically slicing by locations, each analysis contributes only 1/3 of the total information (if there are three locations). The error degrees of freedom will be dramatically reduced and the individual analyses are more powerless as the combined analysis in which the interaction was detected. 2. physically slicing is a lot of work. You have to run many individual analyses which then need to be combined somehow for a meaningful joint interpretation

Slicing without separating the data

The technique we are aiming for does not separate the data but generates the appropriate tests in one analysis. Detection of the significant interaction and its slicing is all done in a single step. This is actually very easy with the newer releases of The SAS System (Version 6.11 and later releases).

Example: An experiment was conducted to assess electrolyte leakage from four different tissue types upon freezing. The experiment was a completely randomized design with a two-factor treatment structure. factor Temperature has eight levels, factor Tssue Type has four levels. The next table shows the two-way (cell) means:

————————————– | | Tissue Type | | |---------------------------| | | 1 | 2 | 3 | 4 | |--------+------+------+------+------| |Tempera-| | | | | |ture | | | | | |--------| | | | | |-8 | 96.8| 81.5| 61.8| 98.3|

|--------+------+------+------+------| |-7 | 87.0| 58.6| 43.6| 96.7|

|--------+------+------+------+------| |-6 | 72.6| 51.7| 33.6| 89.5|

|--------+------+------+------+------| |-5 | 51.5| 33.7| 25.0| 75.4|

|--------+------+------+------+------| |-4 | 36.6| 25.5| 15.9| 64.4|

|--------+------+------+------+------| |-3 | 23.9| 7.4| 14.7| 17.1| |--------+------+------+------+------| |-2 | 17.0| 7.2| 11.8| 15.7| |--------+------+------+------+------| |0 | 4.6| 3.5| 6.5| 4.1| --------------------------------------

With decreasing temperatures leakage increases for any tissue type. The increments are not the same, however.

Plot of LEAK*TEMP. Symbol is value of TISSUE.

100

4 1

2 75

4 1 L e

3 a

2 k 50

1 2 a

3 g

1 e

2 3 25

1 2 3 1 4 3 3 3 1 2 2

0 ----------------------------------------------------- 0 -1 -2 -3 -4 -5 -6 -7 -8

Temperature

NOTE: 4 obs hidden.

With decreasing temperature leakage separates more and more among the tissue types. There appears to be an interaction between temperature and tissue types: a comparison of tissue types depends on what temperature it is performed at. At -2 degrees, the tissue types may not be significantly different, at -7 degrees they may.

To verify whether there is an interaction, run a two-way ANOVA

proc glm data=leakage; class tissue temp; model leak = tissue temp tissue*temp; run; quit;

with abridged output:

General Linear Models Procedure

Class Level Information Class Levels Values TISSUE 4 1 2 3 4 TEMP 8 0 -2 -3 -4 -5 -6 -7 -8

Number of observations in data set = 160

Source DF Sum of Squares Mean Square F Value Pr > F Model 31 158585.4375175 5115.6592748 48.70 0.0001 Error 128 13444.5222800 105.0353303 Total 159 172029.9597975

Source DF Type III SS Mean Square F Value Pr > F TISSUE 3 23847.0616725 7949.0205575 75.68 0.0001 TEMP 7 121221.3735775 17317.3390825 164.87 0.0001 TISSUE*TEMP 21 13517.0022675 643.6667746 6.13 0.0001

The interaction is significant at all reasonable Type-I error levels. To compare the tissue types separately for each temperature, slice the interaction by temperatures:

proc glm data=leakage; class tissue temp; model leak = tissue temp tissue*temp; lsmeans tissue*temp / slice=temp; run;

The additional output table this produces is:

TISSUE*TEMP Effect Sliced by TEMP for LEAK

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

0 3 25.533695 8.511232 0.0810 0.9702 -2 3 288.976095 96.325365 0.9171 0.4347 -3 3 696.837335 232.279112 2.2114 0.0899

-4 3 6589.763440 2196.587813 20.9128 0.0001 -5 3 7427.236375 2475.745458 23.5706 0.0001 -6 3 8898.054000 2966.018000 28.2383 0.0001 -7 3 9107.437500 3035.812500 28.9028 0.0001 -8 3 4330.225500 1443.408500 13.7421 0.0001

Each row of this table corresponds to a comparison of tissue types at a certain temperature. At 0 degrees, there is no difference of the tissue types (P=0.9702). At -2 degrees centigrade the p-value has dropped tp 0.4347, but not sufficiently to declare significance. Continuing, one finds that at th 5% significance level at -4 degrees centigrade the tissue types first show significantly different electrolyte leakage.

It is important to note that these analyses are based on a total of 128 error degrees of freedom and a error mean square estimate of 105.03. This is the error from the overall analysis. For example the F statistic for testing tissue effects at -3 degrees is 232.279/105.03 = 2.2114.

If you perform separate analyses by temperatures;

proc sort data=leakage; by temp; run; proc glm data=leakage; class tissue; model leak = tissue; by temp; run; quit;

and list the sum of squares for tissue effects, you get

Temp Source DF Type I SS 0 TISSUE 3 25.53369500 -2 TISSUE 3 288.97609500 -3 TISSUE 3 696.83733500 -4 TISSUE 3 6589.76344000 -5 TISSUE 3 7427.23637500 -6 TISSUE 3 8898.05400000 -7 TISSUE 3 9107.43750000 -8 TISSUE 3 4330.22550000

Notice that the sum of squares for the separate tissue main effects are identical to the slice sum of squares.

Slicing in PROC MIXED

You can slice interactions in PROC GLM or PROC MIXED (which I prefer). The output produced by the SLICE option of PROC MIXED differs slightly from that of PROC GLM. The statements

proc mixed data=leakage; class tissue temp; model leak = tissue temp tissue*temp; lsmeans tissue*temp / slice=temp; run;

produce the Table of Effect Slices:

Tests of Effect Slices

Effect TEMP NDF DDF F Pr > F

TISSUE*TEMP 0 3 128 0.08 0.9702 TISSUE*TEMP -2 3 128 0.92 0.4347 TISSUE*TEMP -3 3 128 2.21 0.0899 TISSUE*TEMP -4 3 128 20.91 0.0001 TISSUE*TEMP -5 3 128 23.57 0.0001 TISSUE*TEMP -6 3 128 28.24 0.0001 TISSUE*TEMP -7 3 128 28.90 0.0001 TISSUE*TEMP -8 3 128 13.74 0.0001

The first column refers to the effect being sliced. The second column to the level at which a comparison is being performed. The second row, for example, refers to a comparison at -2 degree centigrade. Since the Effect being sliced is TISSUE*TEMP the levels being compared are the four tissue types. This is also obvious from the NDF (=numerator degrees of freedom) column (four tissue types – 1 = 3 ndf). Notice that the F- and p-values are identical to those from PROC GLM.

PROC MIXED has one definite advantage over PROC GLM. Its LSMEANS statement permits the very powerful DIFF option. More on this in the next section.

Comparing Cell Means in the Presence of Significant Interactions

If a main effect is significant and not masked by an interaction, we usually proceed with post-ANOVA procedures, comparing marginal means in pairs (either all pairs or all means against a control or some such thing). If interactions are significant, cell means should be compared instead, since the marginal means may not properly represent the treatment differences at the various levels of a factor. Looking at the table of effect slices from the leakage example,

Effect TEMP NDF DDF F Pr > F

one would concentrate on comparing the cell means for temperatures below -4 degrees centigrade, since there appear to be no differences (at the 5% level) for higher temperatures.

Cell means are reasonably compared only as simple effects. A simple effect is a comparison in which all but one factor are fixed at a certain level. For example a comparison likeTemp(-2),Tissue(1) versus Temp(-5),Tissue(4)is not really meaningful since both factors are varied. Valid simple effects in this example could be Temp(-2),Tissue(1) versus Temp(-2),Tissue(4) Temp(-3),Tissue(1) versus Temp(-5),Tissue(1).

In PROC GLM you can request these comparisons with the PDIFF option of the LSMEANS statement. The MEANS statement which is used for main effects multiple comparisons will not work for interactions (rightfully so).

For example,

proc glm data=leakage; class tissue temp; model leak = tissue temp tissue*temp; lsmeans tissue*temp / pdiff; run;

The output produced in addition to the regular ANOVA table is as follows:

General Linear Models Procedure Least Squares Means

TISSUE TEMP LEAK LSMEAN LSMEAN Number

1 0 4.6320000 1 1 -2 17.0000000 2 1 -3 23.8800000 3 1 -4 36.6480000 4 1 -5 51.4500000 5 1 -6 72.6000000 6 1 -7 86.9600000 7 1 -8 96.8000000 8 2 0 3.4780000 9 2 -2 7.2280000 10 2 -3 7.3540000 11 2 -4 25.4800000 12 2 -5 33.7000000 13 2 -6 51.7200000 14 2 -7 58.6200000 15 2 -8 81.4600000 16 3 0 6.5080000 17 3 -2 11.8400000 18 3 -3 14.7400000 19 3 -4 15.9200000 20 3 -5 24.9800000 21 3 -6 33.6400000 22 3 -7 43.5800000 23 3 -8 61.7800000 24 4 0 4.1160000 25 4 -2 15.6740000 26 4 -3 17.0760000 27 4 -4 64.3600000 28 4 -5 75.3800000 29 4 -6 89.5200000 30 4 -7 96.7400000 31 4 -8 98.2600000 32

Pr > |T| H0: LSMEAN(i)=LSMEAN(j)

i/j 1 2 3 4 5 6 7 8 1 . 0.0586 0.0036 0.0001 0.0001 0.0001 0.0001 0.0001 2 0.0586 . 0.2905 0.0029 0.0001 0.0001 0.0001 0.0001 3 0.0036 0.2905 . 0.0510 0.0001 0.0001 0.0001 0.0001 4 0.0001 0.0029 0.0510 . 0.0240 0.0001 0.0001 0.0001 5 0.0001 0.0001 0.0001 0.0240 . 0.0014 0.0001 0.0001 6 0.0001 0.0001 0.0001 0.0001 0.0014 . 0.0285 0.0003 7 0.0001 0.0001 0.0001 0.0001 0.0001 0.0285 . 0.1315 8 0.0001 0.0001 0.0001 0.0001 0.0001 0.0003 0.1315 . 9 0.8590 0.0390 0.0020 0.0001 0.0001 0.0001 0.0001 0.0001 10 0.6895 0.1341 0.0113 0.0001 0.0001 0.0001 0.0001 0.0001 11 0.6752 0.1392 0.0120 0.0001 0.0001 0.0001 0.0001 0.0001 12 0.0016 0.1931 0.8054 0.0873 0.0001 0.0001 0.0001 0.0001 13 0.0001 0.0111 0.1322 0.6500 0.0071 0.0001 0.0001 0.0001 14 0.0001 0.0001 0.0001 0.0216 0.9668 0.0016 0.0001 0.0001 15 0.0001 0.0001 0.0001 0.0009 0.2707 0.0329 0.0001 0.0001 16 0.0001 0.0001 0.0001 0.0001 0.0001 0.1741 0.3977 0.0195 17 0.7727 0.1080 0.0083 0.0001 0.0001 0.0001 0.0001 0.0001 18 0.2682 0.4275 0.0655 0.0002 0.0001 0.0001 0.0001 0.0001 19 0.1214 0.7279 0.1609 0.0010 0.0001 0.0001 0.0001 0.0001 20 0.0840 0.8679 0.2217 0.0017 0.0001 0.0001 0.0001 0.0001 21 0.0021 0.2205 0.8655 0.0742 0.0001 0.0001 0.0001 0.0001 22 0.0001 0.0114 0.1346 0.6434 0.0069 0.0001 0.0001 0.0001 23 0.0001 0.0001 0.0029 0.2869 0.2269 0.0001 0.0001 0.0001 24 0.0001 0.0001 0.0001 0.0002 0.1135 0.0975 0.0002 0.0001 25 0.9367 0.0490 0.0028 0.0001 0.0001 0.0001 0.0001 0.0001 26 0.0909 0.8382 0.2078 0.0015 0.0001 0.0001 0.0001 0.0001 27 0.0571 0.9907 0.2958 0.0031 0.0001 0.0001 0.0001 0.0001 28 0.0001 0.0001 0.0001 0.0001 0.0485 0.2059 0.0007 0.0001 29 0.0001 0.0001 0.0001 0.0001 0.0003 0.6687 0.0764 0.0012 30 0.0001 0.0001 0.0001 0.0001 0.0001 0.0101 0.6935 0.2635 31 0.0001 0.0001 0.0001 0.0001 0.0001 0.0003 0.1338 0.9926 32 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0837 0.8221

Pr > |T| H0: LSMEAN(i)=LSMEAN(j)

i/j 9 10 11 12 13 14 15 16 1 0.8590 0.6895 0.6752 0.0016 0.0001 0.0001 0.0001 0.0001 2 0.0390 0.1341 0.1392 0.1931 0.0111 0.0001 0.0001 0.0001 3 0.0020 0.0113 0.0120 0.8054 0.1322 0.0001 0.0001 0.0001 4 0.0001 0.0001 0.0001 0.0873 0.6500 0.0216 0.0009 0.0001 5 0.0001 0.0001 0.0001 0.0001 0.0071 0.9668 0.2707 0.0001 6 0.0001 0.0001 0.0001 0.0001 0.0001 0.0016 0.0329 0.1741 7 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.3977 8 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0195 9 . 0.5639 0.5509 0.0009 0.0001 0.0001 0.0001 0.0001 10 0.5639 . 0.9845 0.0056 0.0001 0.0001 0.0001 0.0001 11 0.5509 0.9845 . 0.0060 0.0001 0.0001 0.0001 0.0001 12 0.0009 0.0056 0.0060 . 0.2070 0.0001 0.0001 0.0001 13 0.0001 0.0001 0.0001 0.2070 . 0.0063 0.0002 0.0001 14 0.0001 0.0001 0.0001 0.0001 0.0063 . 0.2891 0.0001 15 0.0001 0.0001 0.0001 0.0001 0.0002 0.2891 . 0.0006 16 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0006 . 17 0.6410 0.9117 0.8964 0.0041 0.0001 0.0001 0.0001 0.0001 18 0.1994 0.4781 0.4901 0.0373 0.0010 0.0001 0.0001 0.0001 19 0.0847 0.2486 0.2566 0.1000 0.0041 0.0001 0.0001 0.0001 20 0.0571 0.1823 0.1887 0.1427 0.0070 0.0001 0.0001 0.0001 21 0.0012 0.0070 0.0075 0.9386 0.1809 0.0001 0.0001 0.0001 22 0.0001 0.0001 0.0001 0.2104 0.9926 0.0061 0.0002 0.0001 23 0.0001 0.0001 0.0001 0.0060 0.1299 0.2115 0.0219 0.0001 24 0.0001 0.0001 0.0001 0.0001 0.0001 0.1231 0.6267 0.0029 25 0.9217 0.6320 0.6182 0.0013 0.0001 0.0001 0.0001 0.0001 26 0.0622 0.1949 0.2016 0.1328 0.0062 0.0001 0.0001 0.0001 27 0.0379 0.1311 0.1361 0.1971 0.0115 0.0001 0.0001 0.0001 28 0.0001 0.0001 0.0001 0.0001 0.0001 0.0534 0.3775 0.0094 29 0.0001 0.0001 0.0001 0.0001 0.0001 0.0004 0.0108 0.3500 30 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.2160 31 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0199 32 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0107 and so forth. The total matrix of all comparisons is of dimension 32*32 since there are 8*4=32 total treatments. Most of the comparisons for which this two-dimensional table contains p-values should be ignored, since they do not correspond to simple effects. To locate the comparisons of interest, first locate the numbers for the means in question in the LSMEANS list. then locate the row and column (or vice versa) for this comparison in the table of p-values. For example, to find out which tissue types differ significantly from each other at -4 degrees centigrade, the temperature at which a tissue effect was first found, locate the means

TISSUE TEMP LEAK LSMEAN LSMEAN Number 1 -4 36.6480000 4 2 -4 25.4800000 12 3 -4 15.9200000 20 4 -4 64.3600000 28

in the table above. To test

Ho: at -4 degrees there is no difference between tissue types 1 and 2,locate the p-value in row 4, column 12 or column 12, row 4. The p-value is 0.0873.

Tissue vs Tissue at Temp Row Column p-value

1 2 -4 12 4 0.0873 1 3 -4 20 4 0.0017 1 4 -4 28 4 0.0001 2 3 -4 20 12 0.1427 2 4 -4 28 12 0.0001 3 4 -4 28 20 0.0001 (not shown)

Comparing cell means in PROC MIXED is much more convenient compared to PROC GLM. The reason is that PROC MIXED displays the results in a more user friendly manner. the statements

proc mixed data=leakage; class tissue temp; model leak = tissue temp tissue*temp; lsmeans tissue*temp / diff; run;

accomplish this. Notice that the option is DIFF, not PDIFF. The LSMEANS statement of PROC GLM does not have a DIFF option and the LSMEANS statement of PROC MIXED does not have a PDIFF option. The portion of the output generated by the DIFF option is

Std Effect TISSUE TEMP _TISSUE _TEMP Difference Error DF t Pr > |t|

TISSUE*TEMP 1 0 1 -2 -12.37 6.48 128 -1.91 0.0586 TISSUE*TEMP 1 0 1 -3 -19.25 6.48 128 -2.97 0.0036 TISSUE*TEMP 1 0 1 -4 -32.02 6.48 128 -4.94 0.0001 TISSUE*TEMP 1 0 1 -5 -46.82 6.48 128 -7.22 0.0001 TISSUE*TEMP 1 0 1 -6 -67.97 6.48 128 -10.49 0.0001 TISSUE*TEMP 1 0 1 -7 -82.33 6.48 128 -12.70 0.0001 TISSUE*TEMP 1 0 1 -8 -92.17 6.48 128 -14.22 0.0001 TISSUE*TEMP 1 0 2 0 1.15 6.48 128 0.18 0.8590 TISSUE*TEMP 1 0 2 -2 -2.60 6.48 128 -0.40 0.6895 TISSUE*TEMP 1 0 2 -3 -2.72 6.48 128 -0.42 0.6752 TISSUE*TEMP 1 0 2 -4 -20.85 6.48 128 -3.22 0.0016 TISSUE*TEMP 1 0 2 -5 -29.07 6.48 128 -4.48 0.0001 TISSUE*TEMP 1 0 2 -6 -47.09 6.48 128 -7.26 0.0001 TISSUE*TEMP 1 0 2 -7 -53.99 6.48 128 -8.33 0.0001 TISSUE*TEMP 1 0 2 -8 -76.83 6.48 128 -11.85 0.0001 TISSUE*TEMP 1 0 3 0 -1.88 6.48 128 -0.29 0.7727 TISSUE*TEMP 1 0 3 -2 -7.21 6.48 128 -1.11 0.2682 TISSUE*TEMP 1 0 3 -3 -10.11 6.48 128 -1.56 0.1214 TISSUE*TEMP 1 0 3 -4 -11.29 6.48 128 -1.74 0.0840 TISSUE*TEMP 1 0 3 -5 -20.35 6.48 128 -3.14 0.0021 TISSUE*TEMP 1 0 3 -6 -29.01 6.48 128 -4.48 0.0001 TISSUE*TEMP 1 0 3 -7 -38.95 6.48 128 -6.01 0.0001 TISSUE*TEMP 1 0 3 -8 -57.15 6.48 128 -8.82 0.0001 TISSUE*TEMP 1 0 4 0 0.52 6.48 128 0.08 0.9367 TISSUE*TEMP 1 0 4 -2 -11.04 6.48 128 -1.70 0.0909 TISSUE*TEMP 1 0 4 -3 -12.44 6.48 128 -1.92 0.0571 TISSUE*TEMP 1 0 4 -4 -59.73 6.48 128 -9.21 0.0001 TISSUE*TEMP 1 0 4 -5 -70.75 6.48 128 -10.91 0.0001 TISSUE*TEMP 1 0 4 -6 -84.89 6.48 128 -13.10 0.0001 TISSUE*TEMP 1 0 4 -7 -92.11 6.48 128 -14.21 0.0001 TISSUE*TEMP 1 0 4 -8 -93.63 6.48 128 -14.44 0.0001 TISSUE*TEMP 1 -2 1 -3 -6.88 6.48 128 -1.06 0.2905 TISSUE*TEMP 1 -2 1 -4 -19.65 6.48 128 -3.03 0.0029 TISSUE*TEMP 1 -2 1 -5 -34.45 6.48 128 -5.31 0.0001 TISSUE*TEMP 1 -2 1 -6 -55.60 6.48 128 -8.58 0.0001 TISSUE*TEMP 1 -2 1 -7 -69.96 6.48 128 -10.79 0.0001 TISSUE*TEMP 1 -2 1 -8 -79.80 6.48 128 -12.31 0.0001 TISSUE*TEMP 1 -2 2 0 13.52 6.48 128 2.09 0.0390 TISSUE*TEMP 1 -2 2 -2 9.77 6.48 128 1.51 0.1341 TISSUE*TEMP 1 -2 2 -3 9.65 6.48 128 1.49 0.1392 TISSUE*TEMP 1 -2 2 -4 -8.48 6.48 128 -1.31 0.1931 TISSUE*TEMP 1 -2 2 -5 -16.70 6.48 128 -2.58 0.0111 TISSUE*TEMP 1 -2 2 -6 -34.72 6.48 128 -5.36 0.0001 TISSUE*TEMP 1 -2 2 -7 -41.62 6.48 128 -6.42 0.0001 TISSUE*TEMP 1 -2 2 -8 -64.46 6.48 128 -9.94 0.0001 TISSUE*TEMP 1 -2 3 0 10.49 6.48 128 1.62 0.1080 TISSUE*TEMP 1 -2 3 -2 5.16 6.48 128 0.80 0.4275 TISSUE*TEMP 1 -2 3 -3 2.26 6.48 128 0.35 0.727

and so forth. The columns TISSUE TEMP list the first combination of the two factors, the columns _TISSUE _TEMP the second combination being compared to the first. For example, the second row compares tissue type 1 at temperature 0 to tissue type 1 at temperature -3 degrees. Not only is this table easier to read than the

uninteresting comparisons in this list, which do not correspond to simple effects:

proc mixed data=leakage; class tissue temp; model leak = tissue temp tissue*temp; lsmeans tissue*temp / diff; make 'diffs' out=diff noprint; run; data diff; set diff; if _effect_ = 'TISSUE*TEMP' then do; if temp ne _temp then delete; if (temp+0) > -4 then delete; end; LSD05 = tinv(0.975,_df_)*_se_; drop _effect_ _se_ _df_; run; proc print data=diff label noobs; format _diff_ 6.2 LSD05 5.1; run;

The MAKE statement of PROC MIXED saves the table of mean differences to the data set diff. It also prevents printing of all the differences to the output window through the NOPRINT option. The DATA step after PROC MIXED deletes all those comparisons for which the two temperatures are not the same. It also deletes comparisons at temperatures below -4 degrees, since no significant slices were found there. Also, the LSD at the 5% level is computed. Here is a printout of the data set diff:

TISSUE TEMP _TISSUE _TEMP Difference t Pr > |t| LSD05

1 -4 2 -4 11.17 1.72 0.0873 12.8 1 -4 3 -4 20.73 3.20 0.0017 12.8 1 -4 4 -4 -27.71 -4.28 0.0001 12.8

1 -5 2 -5 17.75 2.74 0.0071 12.8 1 -5 3 -5 26.47 4.08 0.0001 12.8 1 -5 4 -5 -23.93 -3.69 0.0003 12.8

1 -6 2 -6 20.88 3.22 0.0016 12.8 1 -6 3 -6 38.96 6.01 0.0001 12.8 1 -6 4 -6 -16.92 -2.61 0.0101 12.8

1 -7 2 -7 28.34 4.37 0.0001 12.8 1 -7 3 -7 43.38 6.69 0.0001 12.8 1 -7 4 -7 -9.78 -1.51 0.1338 12.8 1 -8 2 -8 15.34 2.37 0.0195 12.8 1 -8 3 -8 35.02 5.40 0.0001 12.8 1 -8 4 -8 -1.46 -0.23 0.8221 12.8 2 -4 3 -4 9.56 1.47 0.1427 12.8 2 -4 4 -4 -38.88 -6.00 0.0001 12.8

2 -5 3 -5 8.72 1.35 0.1809 12.8 2 -5 4 -5 -41.68 -6.43 0.0001 12.8

2 -6 3 -6 18.08 2.79 0.0061 12.8 2 -6 4 -6 -37.80 -5.83 0.0001 12.8

2 -7 3 -7 15.04 2.32 0.0219 12.8 2 -7 4 -7 -38.12 -5.88 0.0001 12.8

2 -8 3 -8 19.68 3.04 0.0029 12.8 2 -8 4 -8 -16.80 -2.59 0.0107 12.8

3 -4 4 -4 -48.44 -7.47 0.0001 12.8

3 -5 4 -5 -50.40 -7.78 0.0001 12.8

3 -6 4 -6 -55.88 -8.62 0.0001 12.8

3 -7 4 -7 -53.16 -8.20 0.0001 12.8

3 -8 4 -8 -36.48 -5.63 0.0001 12.8

Slicing three-way and higher order interactions

Slicing a higher order interaction is essentially no different from slicing a two-way interaction. You should slice, however, in a manner that generates meaningful comparisons. For example, a significant A*B*C interaction can be sliced in the following ways:

by A by B by C by A*B by A*C by B*C Only the last three are really meaningful, since they compare the levels of one factor while holding the other two factors fixed. As a general rule, if you write a slice statement the number of factors in the SLICE= option of the LSMEANS statement should always be one less than the number of factors in the interaction being sliced. For a three-way interaction, the statements are

lsmeans a*b*c / slice=a*b; lsmeans a*b*c / slice=a*c; lsmeans a*b*c / slice=b*c;

There is a shorthand for combining the three statements into one:

lsmeans a*b*c / slice=(a*b a*c b*c);