Using the Tukey-Kramer method to conducted the pairwise comparisons between different levels of QoLCat

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Background: Dataset exam2.sas7dat is the data of an observational study designed to compare the effectiveness of a new drug, which is viewed by the investigators as the active treatment vs. a standard inhaler, for treating asthma. The patient follow-up period begins at the hospitalization at which treatment for asthma is initiated, using either the new drug or an inhaler. The follow-up period (since the initiation of the interventions) is one year. Variables in the study include: 

Severity = An index for asthma severity, with higher values indicating great severity. 

QOL = quality of life index  

Hosp1yr = a binary 0-1 variable indicating the presence of at least one hospitalization during the 1 year follow-up period 

Treatment = a binary 0-1 indicator variable where 1 indicates that a patient received the active experimental drug and 0 indicates the patient used the inhaler (standard of care). 

QOLCat = a 4-level ordered categorical variable, with higher scores representing a higher QOL 

Num_hosp = the number of hospital, emergency department, or urgent care admissions during the 1 year follow-up period. 

Age1 = Age in years 

Male = 0-1 indicator variable for male sex 

Hispanic = 0-1 indicator variable for Hispanic ethnicity (0=non-Hispanic; 1=Hispanic) 

MaternalEd = Maternal education in years 

Urban = 0-1 indicator for residence in an urban area (0=rural, 1= urban) 

FamIncome = Annual family income in $ 

Question 1: Examine is the effect of the QOL category (QoLCat) on severity. Please answer the following questions (25 pts) 

1. Using One-way analysis of variance (ANOVA) model to test if QoLCat is a statistically significant factor on severity. (15 pts) 

Severity
	Sum of Squares	df	Mean Square	F	Sig.
Between Groups	1153.659	3	384.553	35.772	.000
Within Groups	8546.319	795	10.750
Total	9699.977	798

Ho= QoLCat is not statistically significant factor on severity.

H1= QoLCat is a statistically significant factor on severity.

F= 35.772; p-value <0 .000

Alpha = 0.05, H0 can be rejected since the p-value <0.000, suggesting that there is evidence that QoLCat is a statistically significant factor on severity.

(a) What is the degree of freedom for the QoLCat effect? (2 pts) 

 The degree of freedom for the QoLCat effect is 4-1= 3. 

(b) What is the total sum of square (SS) for severity that is explained by QoLCat? (2 pts) 

The total sum of square for severity that is explained by QoLCat is 1153.659.

1153.659+ 8546.319 = 9699.977.

(c)  What is the mean square (MS) for severity that is explained by QoLCat? (2 pts)               

 The mean square (MS) for severity that is explained by QoLCat is 384.553.

(d)  What is the degree of freedom for the error term? (2 pts) 

The degree of freedom of the error term is 799-4= 795.

(e) What is the mean square error (MSE)? (2 pts) 

The mean square error is 10.750.

(f) Conduct a hypothesis test to assess the effect of QoLCat on severity. (5 pts) 

Ho= QoLCat is not statistically significant factor on severity.

H1= QoLCat is a statistically significant factor on severity.

F= 35.772; p-value <0 .000

Alpha = 0.05, H0 can be rejected since the p-value <0.000, suggesting that there is evidence that QoLCat is a statistically significant factor on severity.

2. Obtain the estimated mean of severity for each of the four QoL levels. (4 pts) 

		N	Mean	Std. Deviation	Std. Error	95% Confidence Interval for Mean		Minimum	Maximum	Between- Component Variance
						Lower Bound	Upper Bound
1		112	19.17	2.956	.279	18.62	19.72	12	26
2		312	17.76	3.205	.181	17.41	18.12	4	27
3		272	16.38	3.125	.189	16.01	16.75	9	26
4		103	15.14	4.126	.407	14.33	15.94	5	27
Total		799	17.15	3.486	.123	16.91	17.39	4	27
Model	Fixed Effects			3.279	.116	16.92	17.38
	Random Effects				.793	14.63	19.67			2.018

3. Using the Tukey-Kramer method to conducted the pairwise comparisons between different levels of QoLCat that are adjusted for multiple comparisons. Which pairs have significant difference in the mean of severity? (Hint: The total number of unique pairwise comparisons is 6). (6 pts) 

K=4 because there are four levels of QOLCat. 4*(4-1)/2=6

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