Reference no: EM131426342
How fast can evolution occur in nature? Are evolutionary trajectories unique or predictable? In 1980, a European Union (EU) fly (Drosophila subobscura) was accidentally introduced into North America. In Europe, the fly's wing size systematically varies with latitude, suggesting an evolutionary adaptation. After allowing two decades for the introduced North American flies to spread over the continent, flies were captured and the hypothesis of speedy evolution was examined by comparing the wing sizes at different latitudes between NA and EU flies.
The data are given below:
continent latitude Wing Size
Females Males
na 35.5 901 797
na 37 896 806
na 38.6 906 812
na 40.7 907 807
na 40.9 898 818
na 42.4 893 809
na 45 913 810
na 46.8 915 819
na 48.8 927 800
na 49.8 924 823
na 50.8 930 814
eu 36.4 905 789
eu 39.3 889 803
eu 41.3 915 812
eu 43.4 930 820
eu 45.5 895 808
eu 47.3 926 815
eu 48.5 944 855
eu 50.4 925 842
eu 52.1 920 819
eu 56.1 934 839
1.
Define the variables as follows:
Y = Wing Size
X1 = Latitude
X2 = dummy code for Continent 1 = NA 0 = EU
X3 = dummy code for Sex 1 = M 0 = F
a. Write the full and restricted models which, in a models comparison framework, would evaluate the null hypothesis that latitude - controlling for continent and sex - has a significant relationship with wing size.
b. (+4) How many degrees of freedom would exist for full and restricted models in part a above?
c. Suppose you were to see the following SAS code in your program editor window:
PROC REG;
MODEL Y = X1 X2 X3;
DEMO: Test X2=0, X3=0;
c1) In WORDS, what is the hypothesis being tested in the test statement labeled DEMO?
c2) Write the full and restricted models used to evaluate the DEMO hypothesis.
d. Write out the expected wing size for a female North American fly captured at a latitude of 45 degrees in terms of the model parameters (we don't have numerical estimates yet) from the full model in part A above.
2. To evaluate the speedy adaptional hypothesis, we need to evaluate whether or not the rates of wing change as a function of latitude vary between EU and NA flies. We may do this by including an interaction term X4 - where X4 is the interaction between latitude and continent. In your favorite program, run a multiple regression model - with wing size as the DV - that includes the linear effects of sex, continent, latitude, and the latitude by continent interaction.
Please answer the following questions.
2a. Which of the effects modeled has the most influence on wing size and how do you know this?
2b. What is the value of the multiple correlation for this model and what is it's sign or direction of influence?
2c. What is the F-value for testing whether or not there are different latitude slopes by continent? What is the companying p-value and squared partial correlation?
2d. Write out the full prediction equation for the model with the estimated parameters in place of the coefficients.
2e. What is the estimated numerical value of Root MSE for this analysis? In words, what is the meaning of this number?
2f. Which observation number has the largest residual? What is the predicted value and observed value associated with this observation?
2g. In words, interpret the coefficient for the interaction term in this model.
2h. What is the numerical value of the t-statistic that would result for the interaction term if all of the coefficients were standardized coefficients?
2i. What is the numerical value of E(R) - E(F) = Δfit for the hypothesis that the interaction term does not influence wing size?
2j. What is the numerical value of the expected wing size for a female North American Fly captured at 45 degrees latitude in this sample?
2k. Using parameter estimates from the interaction model fit in for question 2, what is the estimated intercept and latitude slope for female NA flies?
2l. Using parameter estimates from the interaction model fit in for question 2, what is the estimated intercept and latitude slope for female EU flies?
2m. Using parameter estimates from the interaction model fit in for question 2, what is the estimated intercept and latitude slope for male NA flies?
2n. Using parameter estimates from the interaction model fit in for question 2, what is the estimated intercept and latitude slope for male EU flies?
Question 3. Consider a multiple regression in which we have 4 variables: A response variable Y and 3 explanatory variables: x1, x2, and x3.
Use proper notation for all parts of the question. That is, use
r2y1.2 for (squared) partial correlations,
r2y(1.2), for (squared) semi-partial correlations, and
r2y1 for (squared) simple correlations.
R2y.12 for (squared) multiple correlations
a. Write out the squared multiple correlation between Y and x2 and x3 in terms of a sum of squared simple correlations and squared semi-partial correlations.
b. Write out the squared partial correlation between Y and x2 controlling for x1 and x3 in terms of the squared semi-partial correlation between Y and x2 controlling for x1 and x3.
c. Write the squared partial correlation between Y and x3 controlling for x1 and x2 as a function of squared multiple correlations only.
Question 4. As usual, here is output in which most everything has been erased. For each blank you can fill in, you get 1 point credit. The SAS Code is given as MODEL Y = x1x2;
Correlation
Variable x1 x2 y
x1 1.0000 0.3890 ______
x2 0.3890 1.0000 0.2317
y 0.2565 0.2317 1.0000
Analysis of Variance
Sum of Mean
Source DF Squares Square F Value Pr > F
Model ___ _________ _______ 4.58 0.0126
Error 97 _________ _______
Corrected Total ___ _________
Root MSE 1.03178 R-Square ______
Dependent Mean 11.99894 Adj R-Sq ______
Coeff Var 8.59895
Parameter Estimates
Squared
Parameter Standard Standardized Semi-partial
Variable DF Estimate Error t Value Pr > |t| Estimate Corr Type I
Intercept _ 12.00578 0.10538 ______ <.0001 _______ .
x1 _ 0.22421 _______ 1.86 0.0659 0.19599 _______
x2 _ ______ 0.12275 1.48 0.1432 0.15550 _______
Parameter Estimates
Squared
Partial
Variable Corr Type II
Intercept .
x1 0.03445
x2 _______