# ANOVA and Repeated Measures ANOVA Designs

**ANOVA and Repeated Measures ANOVA Designs**

**Use the University Online Resources to find two peer-reviewed articles in which the authors used ANOVA designs and two peer-reviewed articles in which the authors used repeated measures ANOVA designs. Summarize each article and evaluate whether the design used was logical. Explain your reasoning. Do you think that the design influenced the statistical significance observed? Why or why not? Could this influence the validity of the work?**

**Submission Details**

**Support your responses with examples.****Cite any sources in APA format.**Behavior Research Methods & Instrumentation

1980, Vol. 12 (5), 559-561

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Order Paper NowCapability of SPSS subprogram ANDVA to handle repeated-measures and nested designs

DONALD B. HEADLEY United States Army Biomedical Laboratory, Aberdeen

Proving Ground, Maryland 21010

Although subprogram ANOVA of Statistical Package for the Social Sciences (SPSS; Nie, Hull, Jenkins, Steinbrenner, & Bent, 1975) is basically a factorial design program, its computing capabilities may be readily extended to repeated-measures and nested designs. Advantage may be taken of ANOVA’s capability to perform multiple analyses in a single run. The strategy is to request the appropriate analyses to obtain the proper partitioning of sums of squares (SS). The SS for main effects and interactions are obtained in the usual manner in the first analysis. The additional analyses function to divide the residual SS from the first analysis into SS for the various error terms in a given design.

A data record consists of a datum along with its identifying levels of all factors, including a dummy subjects factor (the factors are listed on a VARIABLE LIST card; the ordering of level numbers on the data record is in accordance with the position of its factor name on this control card). The total number of data records is entered on the N OF CASES control card. The major limitation that prevents obtaining a complete analysis with just one ANOVA statement is that the multiplicative value of the number of factor levels in a given analysis must be less than the total number of data records. This restriction prohibits all the indepen- dent variables listed on the VARIABLE LIST control card to be used in the same analysis if there is one datum per cell (subject-treatments combination). With equal numbers of subjects per group, the multiplicative value of levels must be a whole number multiple of N OF CASES.

The procedures for obtaining the proper partitioning of total SS for repeated-measures designs are shown in Table 1. Both simple and mixed designs are illustrated

Requests for reprints should be sent to Donald B. Headley, Behavioral Toxicology Branch, U.S.A. Biomedical Laboratory, Attention: SGRD-UV-RB, Aberdeen Proving Ground, Maryland 21010. The opinions or assertions contained herein are the private views of the author and are not to be construed as official or as reflecting the views of the Army or the Depart- ment of Defense.

(tor examples of these designs, see Myers, 1972; Winer, 1971). The number of analyses to be requested is equal to the number of within-subjects factors plus one. Many terms of the analysis table are obtained by sub- traction (the degrees of freedom for these terms are likewise determined by subtraction). The values placed in parentheses on the ANOVA procedure card are the ranges of the codes used to define levels of factors. If one has multiple entries in each cell (e.g., multiple determinations from a blood sample) and does not want to analyze these values as a separate factor, no additional individual coding is required; each datum is properly identified by its cell coding. The multiple entries per cell contribute to a within-subjects error term and its corresponding degrees of freedom. ANOVA can also compute SS for designs that have unequal (proportionate or disproportionate) numbers of subjects per group. The user must assume, however, that data for a given subject are present in all levels of the within- subjects factors. No control card adjustments from the equal-number case are necessary. The levels for subjects on the ANOVA statement are the number in the largest group. The default computations for the unequal-number case are in accordance with a least-squares solution (Method 2 described in Overall & Spiegel, 1969; for examples of this method, see Kirk, 1968, pp. 276-281; Winer, 1971, pp. 599-603).

ANOVA statements for three nested designs are shown in Table 2 (see Myers, 1972, for examples). Unless a given design has a repeated-measures variable, a subjects factor need not be coded on data records or listed on the VARIABLE LIST or ANOVA cards. The number of groups within treatments and/or subjects within groups need not be equal.

The requesting of specific analyses on a given data set thus allows the ANOVA factorial program to func- tion as a more general-purpose computational aid.

REFERENCES

KIRK, R. E. Experimental design: Procedures for the behavioral sciences. Belmont, Calif: Brooks/Cole, 1968.

MYERS, J. L. Fundamentals of experimental design (2nd ed.). Boston: Allyn & Bacon. 1972.

NIE, N. H., HULL, C. H., JENKINS, J. G., STEINBRENNER, K., & BENT, D. H. SPSS: Statistical package for the social sciences (2nd ed.). New York: McGraw-Hili, 1975.

OVERALL, J. E., & SPIEGEL, D. K. Concerning least squares analysis of experimental data. Psychological Bulletin, 1969,72, 311-322.

WINER, B. J. Statistical principles in experimental design (2nd ed.). New York: McGraw-Hili, 1971.

Copyright 1980 Psychonomic Society, Inc. 559 0005-7878/80/050559-03$00.55/0

560 HEADLEY

Table I Procedure Dud Coding and Output fOJ Repeated-Measures Designs

Number of Variables’ ANOVA Procedure Sums-of-Squares Terms

B W Card Statements from Output F Ratios”

0 DV BY W(1,3)/ (I)R (2)W 2/4 DV BY S(1,5)/ (3)5 (4)WxS=I-3

0 2 DV BY Wl(I,3) W2(1,3)/ (l)R (2)Wl (3)W2 2/6; (4)WlxW2 3/7;

DV BY Wl(I,3) 5(1,5)/ (5)S (6)WlxS 4/8 DV BY W2(1,3) 5(1,5)/ (7)W2xS

(&)WlxW2xS=I-5-6-7 DV BY BO ,2) W(l,3)/ (I)R (2)B (3)W 2/6 ;

(4)BxW 3/7, DV BY S(l,1 0)/ (5)5 4/7

(6)S w/in B=5-2 (7)WxS w/in B=I-6

2 DV BY B(1,2) wiu,n W2(1,3)/ (1)R (2)B (3)Wl 2/ L1; (4)W2 (5)BxWI 3/ u, (6)BxW2 (7)WlxW2 5/Ll; (8)BlxWlxW2 4/14;

DV BY Wl(I,3) S(1,10)/ (9)5 (10)WlxS 6/14; (11)5 w/in B=9-2 7/15 ; (l2)WlxS w/in B=10-5 8/15

DV BY W2(1,3) S(1,10)/ (l3)W2xS (14)W2xS w/in B=13-6 (lS)WlxW2xS w/in B=1

-11-12~14

2 DV BY B1(1,2) B2(l,2) W(l,3)/ (l)R (2)Bl (3)B2 2/10; (4)W (5)BlxB2 3/10; (6)BlxW (7)B2xW S/W; (8)BlxB2xW 4/11;

DV BY 5(1,20)/ (9)5 6/11; (10)S w/in BIB2=9-2- 7/11;

3-5 (ll)WxS w/in BIB2=1-10 8/11

2 2 DV BY B1(1,2) B2(1,2) Wl(1,3) W2(1,3)/ (1)R (2)Bl (3)B2 2/19; (4)BlxB2 (5)Wl (6)W2 3/19; (7)WlxW2 (8)BlxWI 4/19; (9)BlxW2 (10)B2xWI 5/20; (I1)B2xW2 (l2)BlxB2xWI &/20; (l3)BlxB2xW2 10/20; (l4)BlxWlxW2 12/20; (l5)B2xWlxW2 6/22; (l6)BlxB2xWlxW2 9/22;

DV BY WL(1,3) S(1,20)/ (17)S (l8)WlxS 11/22; (L9)S w/in BIB2=17-2 L3/22;

-3-4 1/23; (20)WlxS w/in BIB2= 14/23;

18-8-10-12 15/23; DV BY W2(l,3) 5(1,20)/ (2l)W2xS (22)W2xS 16/23

w/in BIB2=21-9-11-13 (23)WlxW2xS w/in BIB2

=1-19-20-22

Note-B = between-subjects factor. W = within-subjects factor; D V = dependent variable, R = residual; S = subjects. *AIIume two levels per between-subjects factor and five Iii bjects per group; assume three levels per within-subjects factor. **Use mean squares ofindicated terms.

REPEATED MEASURES AND NESTED DESIGNS FROM SPSS’s ANOVA 561

Table 2 Procedure Card Coding and Output for Nested Designs

Design

Groups within treatments*

Groups within treatments with a variable within groups**

Groups within treatments with a variable within groups and repeated measures within this variablej

ANOVA Procedure Card Statements

DV BY T(l,2)1 DV BY G(l,6)1

DV BY TO ,2) W10 ,2)1 DV BY GO ,6) W10 ,2)1

DV BY TO,2) W10 ,2) W20,3)1

DV BY G(l,6) W1(l,2) W2(l,3)1

DV BY S(1,60)1

Sums-of-Squares Terms from Output

(l)R (2)T (3)G (4)G w/in T=3-2 (5)S wlin G w/in T=1-4 (1)R (2)T (3)W1 (4)TxWl (5)G (6)GxWl (7)G w/in T=5-2 (8)GxWI w/in T=6-4 (9)S wlin GxWI w/in T=I-7-8 (l)R (2)T (3)W1 (4)W2 (5)TxWl (6)TxW2 (7)W1xW2 (8)TxW1xW2 (9)G (l O)GxWI Ol)GxW2 (l2)GxW1xW2 (l3)G w/in T=9-2 (l4)GxWl wlin T=10-5 (l5)GxW2 w/in T=11-6 (16)GxW1xW2 w/in T=12-8 (17)S (18)S w/in GxWl wlin

T=17-9-3-5-14 (19)SxW2 w/in GxWl w/in T=

1-13-14-15-16-18

F Ratiostt

2/4; 4/5

2/7; 3/8; 4/8; 8/9

2/13; 3/14; 5/14; 14/18; 4115; 6/15; 15/16; 7/16; 8/16; 16/19

Note-DV = dependent variable, R = residual; G = groups, S = subjects, T = treatments; Wi = within-groups variable, W2 x: within- subjects variable. *Assume two levels for treatments and three groups per treatment level. **Assume two levels for the variable within groups. [Assume three levels for the within-subjects variableand five subjects per group. ttUse mean squares of indicated terms.

(Accepted for publication July 2, 1980.)