Question

The Novaco Provocation Inventory (NPI) is an instrument for assessing anger responsiveness. The test consists of brief descriptions of situations of provocation, for which the respondent notes the degree of anger that he/she might experience if that event should occur in his/her life. The ratings are on a 5-point Likert-type scale. The main purpose of the NPl is to gauge the range and intensity of a person's anger. Three groups of inmates from a state prison were selected and grouped on the basis of whether this was a first, second, or a third incarceration. Each man took the NPl and the results were as follows: (Be careful coding and entering the data)

    First        Second           Third

223	347	397
202	319	310
234	320	335
266	330	320
292	290	315
237	337	310
176	310	337
229	290	300
225	295	330
147	327	310
149	290	300
210	320	317
319	335	320
278	290	300
266	310	247
176	287	310

1. Write out the null hypothesis using appropriate IV and DV terms – 5 points.

2. Report the means and SD for each group– 5 points.

3. Report all results for the Levene’s Test of Homogeneity of variances and explain – 5 points.

4. Complete the Source table below– 10 points:

Source of Variation	Sums of Squares	Degrees of Freedom	Mean Square	F	p
Incarceration
Within
Total

4. Do you accept or reject the null hypothesis? (select one) and on what basis– 5 points?

5. Report the results of the Tukey HSD Post Hoc test– 5 points.

6. If you reject the null, using the Tukey HSD results, identify which group(s) differed significantly from each other – 5 points?

7. Write up a summary explanation of the overall results– 10 points.

Note. This is a One-Way ANOVA (i.e., 1 IV). The one IV (factor) has three levels and there is one DV (score on the test).

Answer 1

The Novaco Provocation Inventory (NPI) is either an 80 or 90 item instrument for assessing anger responsiveness. (in order to maximize the variability, the 90-item version was used in this study) The test consists of brief descriptions of situations of provocation, for which the respondent notes the degree of anger that he/she might experience if that event should occur in his/her life. The ratings are on a 5-point Likert-type scale. The main purpose of the NPl is to gauge the range and intensity of a person's anger. Three groups of inmates from a state prison were selected and grouped on the basis of whether this was a first incarceration, a second incarceration, or a third incarceration. Each man took the NPl and the results were as follows:

There are three groups.

For three groups we use one way ANOVA.

Descriptive statistics of your k=3 independent treatments:

Treatment →	A	B	C	Pooled Total
observations N	19	19	19	57
sum ∑xi	4,311.0000	5,922.0000	5,828.0000	16,061.0000
mean x¯	226.8947	311.6842	306.7368	281.7719
sum of squares ∑x2i	1,016,289.0000	1,851,912.0000	1,795,126.0000	4,663,327.0000
sample variance s2	2,119.2105	339.8947	414.6491	2,460.5006
sample std. dev. s	46.0349	18.4362	20.3629	49.6034
std. dev. of mean SEx¯	10.5611	4.2296	4.6716	6.5701

One-way ANOVA of your k=3 independent treatments:

source	sum of squares SS	degrees of freedom νν	mean square MS	F statistic	p-value
treatment	86,060.4561	2	43,030.2281	44.9206	3.2493e-12
error	51,727.5789	54	957.9181
total	137,788.0351	56

Conclusion from Anova:

The p-value corresponing to the F-statistic of one-way ANOVA is lower than 0.05, suggesting that the one or more treatments are significantly different. The Tukey HSD test follow. These post-hoc tests would likely identify which of the pairs of treatments are significantly differerent from each other.

Tukey HSD Test:

The p-value corrresponing to the F-statistic of one-way ANOVA is lower than 0.01 which strongly suggests that one or more pairs of treatments are significantly different. You have k=3 treatments, for which we shall apply Tukey's HSD test to each of the 3 pairs to pinpoint which of them exhibits statistially significant difference.

We first establish the critical value of the Tukey-Kramer HSD Q statistic based on the k=3 treatments and ν=54 degrees of freedom for the error term, for significance level α= 0.01 and 0.05 (p-values) in the Studentized Range distribution. We obtain these ctitical values for Q, for α of 0.01 and 0.05

as Qcritical(α=0.01, k=3, v=54) = 4.3004

Qcritical(α=0.05, k=3, v=54) = 3.4082 respectively.

We calculate a parameter for each pair of columns being compared, which we loosely call here as the Tukey-Kramer HSD Q-statistic, or simply the Tukey HSD Q-statistic, as:

Qi,j = |xibar - xjbar | / si,j

where the denominator in the above expression is:

si,j = σ^ϵ / sqrt(Hi,j)   i,j=1,…,k; i≠j.

The quantity Hi,j is the harmonic mean of the number of observations in columns labeled i and j. Note that when the sample sizes in the columns are equal, then their harmonic mean is simply the common sample size. When the sample sizes of columns in a pair being compared are different, the harmonic mean lies somewhere in-between the two sample sizes. The relvant harmonic mean is required for applying the Tukey-Kramer procedure for columns with unequal sample sizes.

The quantity σ^ϵ = 30.9503 is the square root of the Mean Square Error = 957.9181 determined in the precursor one-way ANOVA procedure.

Note that σ^ϵ is same across all pairs being compared. The only factor that varies across pairs in the computation of si,j=σ^ϵsqrt(Hi,j) is the denominator, which is the harmonic mean of the sample sizes being compared.

The test of whether the NIST Tukey-Kramer confidence interval includes zero is equivalent to evaluating whether Qi,j>Qcritical, the latter determined according to the desired level of significance α (p-value), the number of treatments k and the degrees of freedom for error ν, as described above.

post-hoc Tukey HSD Test Calculator results:

k=3 treatments
degrees of freedom for the error term ν=54
Critical values of the Studentized Range Q statistic:

Qcritical (0.01) = 4.3004

Qcritical (0.05) = 3.4082

Green colour values are significant while red colour value is insignificant.

Tukey HSD results

treatments pair	Tukey HSD Q statistic	Tukey HSD p-value	Tukey HSD inferfence
A vs B	11.9414	0.0010053	** p<0.01
A vs C	11.2446	0.0010053	** p<0.01
B vs C	0.6968	0.8632537	insignificant