Question

[1] An article in Bio-electromagnetics (Electromagnetic Effects on Forearm Disuse Osteopenia: A Randomized, Double-Blind, Sham-Controlled Study," Vol. 32. 2011. pp. 273-282) described a randomized, double-blind, sham-controlled, feasibility and dosing study to determine if a common pulsing electromagnetic field (PEMF) treatment could moderate the substantial osteopenia that occurs after forearm disuse. Subjects were randomized into four groups after a distal radius fracture, or carpal surgery requiring immobilization in a cast. Active or identical sham PEMF transducers were wom on the distal forearm for 1, 2, or 4h day for 8 weeks starting after cast removal (baseline") when bone density continues to decline. Bone mineral density (BMD) and bone geometry were measured in the distal forearm by dual energy X-ray absorptiometry (DXA) and peripheral quantitative computed tomography (PQCT). The data below are the percent losses in BMD measurements on the radius after 16 weeks for patients wearing the active or sham PEMF transducers for 1.2, or 4h day (data were constructed to match the means and standard deviations read from a graph in the paper). a) Is there evidence to support a claim that PEMF usage affects BMD loss? If so, analyze the data to determine which specific treatments produce the differences b) Analyze the residuals from this experiment and comment on the underlying assumptions and model adequacy. Sham 4.51 7.95 1 h per day 5.32 4.97 PEMF 2 h per day 4.73 5.81 5.69 3.86 4.06 6.56 8.34 3.01 6.71 3 h per day 7.03 4.65 6,65 5.49 6.98 4.85 7.26 5.92 5.58 6.51 7.91 4.9 3 7.97 2.23 3.95 5.64 9.35 6.52 4.96 6.1 7.19 4.03 2.72 9.19 5.17 5.7 5.85 6.45 5.12 7.08 5.48 6.52 4.09 6.28 7.77 5.68 8.47 4.58 4.11 5.72 5.91 6.89 6.99 4.98 9.94 6.38 5.89 6.55 5.34 5.88 7.5 3.28 5.38 4.54 8.18 5.42 6.03 7.04 5.17 7.6 7.9 7.91 7.3 5.46 [2] A product developer is investigating the tensile strength of a new synthetic fiber that will be used to make cloth for men's shirts. Strength is usually affected by the percentage of cotton used in the blend of materials for the fiber. The engineer conducts a completely randomized experiment with five levels of cotton content and replicates the experiment five times. The data are shown in the following table. a) Is there evidence to support the claim that cotton content affects the mean tensile strength? Use -0.05. b) Use the Fisher LSD method to make comparisons between the pairs of means. What conclusions can you draw?

It's a question from design of Experiments

Answer 1

Question 1:

(a) The hypothesis being tested is:

H₀: µ₁ = µ₂ = µ₃ = µ₄

H_a: At least one means is not equal

The ANOVA output is:

Anova: Single Factor
SUMMARY
Groups	Count	Sum	Average	Variance
Sham	20	113.45	5.6725	4.009693
PEMF 1h/day	20	123.31	6.1655	2.084973
PEMF 2h/day	20	109.56	5.478	2.705322
PEMF 4h/day	20	127.01	6.3505	1.517384
ANOVA
Source of Variation	SS	df	MS	F	P-value	F crit
Between Groups	10.0435	3	3.347835	1.297941	0.281326	2.724944
Within Groups	196.0301	76	2.579343
Total	206.0736	79

The p-value is 0.281326.

Since the p-value (0.281326) is greater than the significance level (0.05), we cannot reject the null hypothesis.

Therefore, we cannot conclude that PEMF usage affects BMD loss.

(b) The residuals plots are:

The assumptions of homogeneity of variance is not met from the residual plot.