We are examining the results of a study of eye-witness testimony. The investigat

Question

We are examining the results of a study of eye-witness testimony. The investigators wish to know whether recall of important information is influenced by the severity of the crime and the amount of time that passes before the witness is asked to recall important events. The study is described below. The first independent variable is severity of the crime. A videotape is made of a man stealing a woman's purse in an uncrowded clothing store. A second videotape is made, using the same actors, of a man using a gun to rob the cash register of the same clothing store. In this scenario, the woman is a sales clerk behind the cash register and there are two people standing at the counter. The second independent variable is time to recall. Subjects are asked to view the videotape and to imagine that they are at the crime scene. In the "immediate" condition, subjects are asked a series of questions about the crime within 10 minutes of viewing the videotape. In the "delay" condition, they are asked the same series of questions about the crime 20-30 minutes after viewing the tape. This is designed to be similar to an actual crime scene where there is some delay when taking witness statements. Perform the appropriate analysis by hand. Please write up the results as you would in a scientific paper. For any significant main effects, please report the effect site. sigma X^2 = 1, 461 G = 233 G^2 = 54, 289

Explanation / Answer

Solution

A two-way ANOVA with equal number of observations per cell would be the analysis tool,

treating severity of crime (Armed robbery or Purze snatching) as row effect and time lapse between the event and recall (Immediate or Delay )as column effect.

Back-up Theory

Suppose we have data of a 2-way classification ANOVA, with r rows, c columns and n observations per cell.

Let xijk represent the kth observation in the ith row-jth column, k = 1,2,…,n; i = 1,2,……,r ; j = 1,2,…..,c.

Then the ANOVA model is: xijk = µ + i + j + ij + ijk, where µ = common effect, i = effect of ith row, j = effect of jth column, ij = row-column interaction and ijk is the error component which is assumed to be Normally Distributed with mean 0 and variance 2.

Now, to work out the solution,

Terminology:

Cell total = xij. = sum over k of xijk

Row total = xi..= sum over j of xij.

Column total = x.j. = sum over i of xij.

Grand total = G = sum over i of xi.. = sum over j of x.j.

Correction Factor = C = G2/N, where N = total number of observations = r x c x n =

Total Sum of Squares: SST = (sum over i,j and k of xijk2) – C

Row Sum of Squares: SSR = {(sum over i of xi..2)/(cxn)} – C

Column Sum of Squares: SSC = {(sum over j of x.j.2)/(rxn)} – C

Between Sum of Squares: SSB = {(sum over i and jof xij.2)/n} – C

Interaction Sum of Squares: SSI = SSB – SSR – SSC

Error Sum of Squares: SSE = SST – SSB

Mean Sum of Squares = Sum of squares/Degrees of Freedom

Degrees of Freedom:

Total: N (i.e., rcn) – 1;

Between: rc – 1;

Within(Error): DF for Total – DF for Between;

Rows: (r - 1);

Columns: (c - 1);

Interaction: DF for Between – DF for Rows – DF for Columns;

Fobs:

for Rows: MSSR/MSSE;

for Columns: MSSC/MSSE;

for Interaction: MSSI/MSSE

Fcrit: upper % point of F-Distribution with degrees of freedom n1 and n2, where n1 is the DF for the numerator MSS and n2 is the DF for the denominator MSS of Fobs

Significance: Fobs is significant if Fobs > Fcrit

Calculations:

We have r = 2, c = 2, n = 10, N = 40. x2 = 1461, x11. = 63, x12. = 38, x21. = 77, x22. = 55,

G = 233 and hence C = 1357.225

SST = 1461 – 1357.225 = 106.775 = SST

SSB = {(632 + 382 + 772 + 552)/10} – C = 1436.7 – 1357.225 = 79.475 = SSB

SSR = [{(63 + 38)2/20} + {(77 + 55)2/20}] – C = 1381.25 – 1357.225 = 24.025 = SSR

SSC = [{(63 + 77)2/20} + {(38 + 55)2/20}] – C = 1412.45 – 1357.225 = 55.23 = SSC

SSW(SSE) = SST – SSB = 106.775 – 79.475 = 27.3 = SSE

SSI = SSB – SSR – SSC = 79.475 – 24.025 – 55.23 = 0.220 = SSI

ANOVA TABLE [level of significance is taken to be 5%]

Source

DF

SS

MS

FCAL

FCRIT

Significance

Row

1

24.025

24.025

34.321

4.113

Significant

Column

1

55.23

55.23

78.9

4.113

Significant

Interaction

1

0.220

0.220

0.31

4.113

Not significant   

Between

3

79.475

-

Error

36

27.3

0.7

Total

39

106.75

-

Conclusion

Both severity of crime and time lapse between event and recall have significant effect on the response of the witnesses.

DONE

Source

DF

SS

MS

FCAL

FCRIT

Significance

Row

1

24.025

24.025

34.321

4.113

Significant

Column

1

55.23

55.23

78.9

4.113

Significant

Interaction

1

0.220

0.220

0.31

4.113

Not significant   

Between

3

79.475

-

Error

36

27.3

0.7

Total

39

106.75

-

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