Statistics 252 - Homework # 4 - Fall 2015 STAT 252 Homework 4 (50 marks) - Due Friday, November 27 at 10pm Show all your work and include appropriate justification to receive full credit for a solution. Single numerical answers (unless indicated) will not receive any marks. In particular: Whenever you are asked to \"carry out the most appropriate test\": i) ii) iii) iv) v) Define the parameter(s) of interest. State clearly the null and alternative hypothesis in terms of the parameter(s). Calculate the test statistic and state all of its components. T-test: State the estimate, hypothesized value and standard error. F-test: State the sum-of-squared residuals and degrees of freedom for each model. Calculate the p-value for the test. Give a probability statement identifying how the p-value is calculated. That is, show how the test statistic is compared to the null distribution of the test statistic. If the p-value is given in output, then it is fine to report it as is. If it is not, then you must estimate the p-value (within a range of values), using the appropriate table. Answer the question in plain English by referring to the p-value (and a significance level if given). Whenever you are asked for a confidence interval: i) ii) iii) iv) v) State the estimate. State the standard error. State the critical value. Calculate the interval. Interpret the interval. Other: If you need to use the t-table and the t-distribution you need is NOT on the table, round your degrees of freedom DOWN to the nearest one. 1 Statistics 252 - Homework # 4 - Fall 2015 1. (30 marks total) Persistent patent ductus arteriosus (PDA) is a medical condition that affects many very low-birth weight infants. Infants with PDA have an increased risk of serious ailment such as severe heart failure. The Apgar score is used to assess the health of an infant one minute after birth. The one-minute Apgar score measures how well the newborn tolerated the birthing process. A score of seven to 10 is normal and indicates the newborn is in good condition. Any score less than seven indicates that the baby needs assistance such as helping the infant breathe. Airway pressure is a measure of respiratory status, usually measured seven days after birth. Some researchers determined that early ibuprofen administration might prevent PDA in preterm neonates. A sample of 130 such infants was randomly divided into 4 groups. 32 were treated with one dose of ibuprofen, 33 with two doses, 33 with three doses, and 32 with a saline solution. The following is the description of variables under study. Variable Gender Dose Weight Age Score Pressure Description of Variables Male OR Female Dose of ibuprofen (an integer from 0 to 3) Birth weight (in pounds) Gestational age (in weeks) One-minute Apgar score Air-way pressure (in cm H 2 O ) seven days after birth For parts (a) - (d): (No output is required for these parts) Consider the regression model below (which will be referred to as the \"original model\" for parts (a) - (d)) for average air-way pressure seven days after birth given the variables, weight, gender, and dose (where gender and dose are treated as categorical variables and weight is a numerical variable). ( pressure | weight , GENDER, DOSE ) 0 1weight 2 male 3 D1 4 D 2 5 D3 6 ( weight male) 7 ( weight D1) 8 ( weight D 2) 9 ( weight D3) 10 ( weight male D1) 11 ( weight male D 2) 12 ( weight male D3) The indicator variables for gender and dose are defined as follows: 1, for a male 0, for a female GENDER: male DOSE: 1, if dose is 1 D1 0, otherwise 1, if dose is 2 D2 0, otherwise 1, if dose is 3 D3 0, otherwise 2 Statistics 252 - Homework # 4 - Fall 2015 a) (6 marks: 2 marks for the general effect, and then 0.5 for each individual one ) Referring to the original model, in terms of the regression coefficients, what is the effect of weight on mean pressure? Find this effect in general, and then summarize it for each combination of gender and dose of ibuprofen. Summarize your results in a chart like the one below. Gender Dose of ibuprofen Male 0 Male 1 Male 2 Male 3 Female 0 Female 1 Female 2 Female Effect of weight on mean pressure 3 b) (3 marks) Modify the original model to specify that the effect of weight on the mean of pressure is the same for males and females with the same dose of ibuprofen; otherwise, the effect of weight on the mean of pressure is possibly different for males and females with different doses of ibuprofen. Just state the constraint(s) needed. You do not have to rewrite the model. c) (4 marks) Referring to the original model, set up a test to explore whether the effect of weight is any different for the different doses of ibuprofen for females. Write out the null and alternative hypothesis in terms of the regression coefficients, and identify the distribution of the test statistic under the null hypothesis. d) (4 marks: 2 marks for the general effect, and then 0.5 for each individual one.) Referring to the original model, in terms of the regression coefficients, what is the effect of gender (male vs. female) on the mean of pressure? Find this effect in general, and then summarize it for each dose of ibuprofen. Summarize your results in a chart like the one below. Dose of ibuprofen Effect of gender (male vs. female) on the mean pressure 0 1 2 3 3 Statistics 252 - Homework # 4 - Fall 2015 e) (7 marks) Consider the following three models: Model 1: {Pressure| Dose} 0 1 D1 2 D2 3 D3 Model 2: {Pressure | Weight , Age, GENDER} 0 4Weight 5 Age 6 Male , Model 3: {Pressure | DOSE ,Weight , Age, GENDER} 0 1 D1 2 D2 3 D3 4Weight 5 Age 6 Male . Using the output below, carry out the most appropriate test to determine if there are any differences in mean pressure for the four different levels of dose, after taking into account weight, age, and gender. Use a significance level of 0.05. ANOVA table for model 1: ANOVA table for model 2: ANOVA table for model 3: f) (6 marks) For each of the models defined above, calculate R-squared and the adjusted R-squared value. 4 Statistics 252 - Homework # 4 - Fall 2015 2. (20 Marks in Total) An agronomist studied the effects of moisture (in inches) and temperature (in 0 C ) on the yield of a new hybrid tomato. She chose five levels for moisture (6, 8, 10, 12, and 14) and five levels for temperature (20, 21, 22, 23, and 24), conducting an experiment with two replications for each combination of moisture and temperature. The following table gives some statistical summaries of data. Variable Sample Mean Sample S.D. Yield 47.097 7.07121 Moisture 10.000 2.85714 Temperature 22.000 1.42857 a) (7 marks) To explore the effects of moisture and temperature, the researchers decided to use a TwoWay ANOVA. Does it appear that the effects of moisture on the mean yield depend on temperature? Carry out the most appropriate test using a significance level of 0.01. Regardless of your answer in part (a), use the information below, showing the results of the additive twofactor ANOVA for the effects of moisture and temperature on mean yield, to answer parts (b) - (d). For each part, you do NOT need to show all the steps of a hypothesis test; just identify the models under the null and alternative hypothesis, state the test statistic (no calculation required), and the p-value probability statement with the p-value. b) (3 marks) Is there any evidence that either moisture or temperature have an effect on yield? c) (3 marks) Is there any evidence that moisture has an effect on yield, after accounting for temperature? d) (3 marks) Is there any evidence that temperature has an effect on yield, after accounting for moisture? e) (4 marks total) Given the information above, find the sum-of-squared residuals for the following models. i) (2 marks) The One Factor model for only moisture. ii) (2 marks) The One Factor model for only temperature. 5