Question

PLEASE SOLVE USING RSTUDIO

1. Eighteen freshwater clams were randomly assigned to 3 groups of 6 each. One group was placed in the pond water from which the clams were collected, one group was placed in deionized water, and one group was placed in a solution of 0.5 mM sodium sulfate. At the end of the specified time period, blood potassium levels were determined.

i. Consider and comment on the technical assumptions of ANOVA. Conduct statistical tests where possible and include relevant results. Do you think ANOVA is appropriate? [10 pts]

ii. If you think it is appropriate to do ANOVA: Conduct an ANOVA to determine whether these treatments affect blood potassium levels (μM K+). Present a complete (with total line) ANOVA table and interpret. If you think it isn’t inappropriate to do ANOVA: Conduct a Kruskal-Wallis test to determine whether these treatments affect blood potassium levels (μM K+). [10 pts]

iii. What follow-up procedure might you suggest after conducting the basic test? [5 pts]

Pond Water	Deionized	Sodium Sulfate
0.518	0.318	0.393
0.523	0.342	0.415
0.495	0.301	0.351
0.502	0.390	0.390
0.525	0.327	0.385
0.490	0.320	0.397

Answer 1

(i)

Assumption of ANOVA:

1. The experimental errors of the observations are normally distributed

2. Equal variances between treatments (i.e. assume Homoscedasticity)

3. Independence of samples

Test for Equal Variances: blood potassium levels versus Group

95% Bonferroni confidence intervals for standard deviations

Group N Lower StDev Upper
Deionized 6 0.0175531 0.0309322 0.0966365
Pond Water 6 0.0085642 0.0150919 0.0471493
Sodium Sulfate 6 0.0119452 0.0210499 0.0657630

Bartlett's Test (Normal Distribution)
Test statistic = 2.31, p-value = 0.315

Levene's Test (Any Continuous Distribution)
Test statistic = 0.33, p-value = 0.725
From the above test we can assume the variances of each groups are equal.

From Kolmogrov-Smirnov test it is observed that assumption of normality is also hold.

Hence we can perform ANOVA.

(ii)

One-way ANOVA: blood potassium levels versus Group

Source DF SS MS F P
Group 2 0.096955 0.048478 89.35 0.000
Error 15 0.008138 0.000543
Total 17 0.105094

From the above ANOVA table, we observe that p-value<0.000 so we conclude that  these treatments affect blood potassium levels significantly and effects of three treatments are not all same.

(iii)

Now we need to perform post hoc test. Here we use Tukey's test.

Tukey 95% Simultaneous Confidence Intervals
All Pairwise Comparisons among Levels of Group

Individual confidence level = 97.97%

Group = Deionized subtracted from:

Group Lower Center Upper
Pond Water 0.14093 0.17583 0.21073
Sodium Sulfate 0.02060 0.05550 0.09040

Group ------+---------+---------+---------+---
Pond Water (---*--)
Sodium Sulfate (---*--)
------+---------+---------+---------+---
-0.10 0.00 0.10 0.20

Group = Pond Water subtracted from:

Group Lower Center Upper
Sodium Sulfate -0.15523 -0.12033 -0.08543

Group ------+---------+---------+---------+---
Sodium Sulfate (---*--)
------+---------+---------+---------+---
-0.10 0.00 0.10 0.20

From the above confidence intervals we observe that none of these confidence intervals contains zero hence all three groups are different.