4 MULTIPLE LINEAR REGRESSION In this assignment, you will use multiple regression tools available in SPSS to examine the relationship between extinction time of land-bird species and average number of nesting pairs. You will compare different models to choose the one that adequately approximates the mean of the response as a function of the explanatory variables and conveniently allows for the questions of interest to be investigated. In particular, you will evaluate the evidence of an association between species size or migratory status and extinction time after accounting for the number of nesting pairs. Moreover, you will test the regression model assumptions and apply diagnostic tools in SPSS. Extinction Time of Land-Bird Species A group of scientists expects that species with larger numbers of nesting pairs will tend to remain longer before becoming extinct. In order to verify the claim, they obtained measurements on breeding pairs of land-bird species collected from 16 islands around Britain over the course of several decades. (S.L. Pimm, H.L. Jones, and J. Diamond, \"On the Risk of Extinction,\" American Naturalist 132 (1988): 757-85.). The response variable is time of extinction on the island where the birds appeared and the explanatory variables are number of nesting pairs, the size of the species, and the migratory status of the species. The data are available in the SPSS file lab4.sav that can be downloaded to your local station from the Statistical Laboratories web site at http://www.stat.ualberta.ca/statslabs/index.htm (click Stat 252 link and Data for Lab 4). The data are not to be printed in your submission. The following is a description of the variables in the data file: Column 1 2 3 Variable Name Species Time Pairs 4 5 Size Status Description of Variable Name of species Average time of extinction on those islands where species appeared The average number of nesting pairs (the average, over all islands where the birds appeared, of the number of nesting pairs per year) Size of the species (categorized as L or S: L for Large and S for small) Migratory status of species (categorized as R or M: R for resident and M for migrant) Answer the following questions using the data: 1. First you will use the Scatterplot tool in SPSS to verify if there is any association between time of extinction and number of pairs. (a) Obtain a scatterplot of time versus number of pairs using different markers for each species size. Paste the plot into your report. Comment briefly on the pattern of association between time of extinction and number of pairs. In particular, does the data support the notion that species with larger numbers of nesting pairs will tend to remain longer before becoming extinct? Explain briefly. Is the pattern of association the same for each species size? Is there any observable outliers? If any, what is (are) the name(s) of this (these) outlying species? (b) Regardless of species size and migratory status, does the data present a homogeneous association between time to extinction and number of pairs? 1 2. Use the Correlate tool in the SPSS Analyze menu to obtain the Pearson's correlation coefficient between time of extinction and number of pairs. Report this coefficient and comment on its sign and magnitude. Are the sign and magnitude of the association consistent with your observations from the scatterplot of question 1, part (a)? 3. Now obtain a natural logarithm transformation of time of extinction, or ln(time). 4. 5. 6. (a) Obtain and paste a scatterplot of ln(time) versus number of pairs (there is no need to set markers for any variable here). Comment on the pattern of association between ln(time) and number of pairs (direction and strength). Moreover, compare the association on the logtransformed and the original scale of measurement. (b) Report the Pearson's correlation coefficient between the ln(time) and number of pairs. Comment on the magnitude of this correlation, making a specific comparison to the untransformed scale. Now you will use the regression tool in SPSS to study the influence of number of pairs, species status, and migratory status on ln(time). Paste relevant SPSS output into your report as needed. (a) Define a multiple linear regression model with the log-transformed time of extinction as the response and number of pairs, size of species, and migratory status of species as explanatory variables. State the assumptions necessary for the validity of this model. (b) What is the estimated regression model of ln(time) on number of pairs, size of species, and migratory status of species? (c) Test the significance of the overall model. State this question as null and alternative hypotheses about the regression coefficients. Report the value of the test statistic, the distribution of the test statistic under the null hypothesis, and the p-value. State your conclusions. (d) What is the percentage of the variation in ln(time) explained by the explanatory variables in the model? (e) Comment on the significance of each explanatory variable individually, given the other variables in the model. Now you will investigate the adequacy of the model you fitted in Question 4. (a) Obtain a plot of standardized residuals versus standardized predicted values. Paste the plot into your report. Is there any evidence that the constant variance assumption is violated? Any outliers? Explain briefly. (b) Obtain a normal probability plot of standardized residuals. Paste the plot into your report. Is there any evidence that the normality assumption is violated? (c) Obtain the case statistics (in particular, leverages, studentized residuals, and Cook's distances) for the 62 species. Examine these statistics carefully for each species and identify outliers and potential influential cases that may be present in this data set. Pay special attention to the case(s) that had been identified as outliers in part (a). Remove cases 61 and 62 from the data. (a) Explain why the two cases may have a large effect on the regression line. Refer to the plots and case statistics obtained in Question 5 in your explanations. 2 Rerun the regression model of Question 4 with the new dataset. Paste relevant outputs into your report (relevant means those that you need to answer the questions below). (b) What is the estimated regression model of ln(time) on number of pairs, size of species, and migratory status of species? (c) What percentage of the variation in ln(time) is now explained by the model? Compare the value with the value reported in part (d) of Question 4. (d) Obtain a plot of standardized residuals versus standardized predicted values. Paste the plot into your report. Is there any evidence that the constant variance assumption is violated? (e) Obtain a normal probability plot of standardized residuals. Paste the plot into your report. Is there any evidence that the normality assumption is violated? 7. After accounting for the number of nesting pairs, do species size or migratory status have an effect on ln(time)? In order to address this question, fit a regression model with ln(time) as the response variable and number of pairs as the only explanatory variable. State the above question as null and alternative hypotheses about the appropriate regression coefficients. Use appropriate SPSS outputs and hand calculations to obtain the value of a test statistic, state the distribution of the test statistic under the null hypothesis, and determine the corresponding p-value of the test. State your conclusions. 8. Does the effect of species size depend on the number of nesting pairs, after accounting for migratory status? Investigate this question by deriving the appropriate interaction variable. Add this variable to the model used in Question 6. Write out the new model. State the above question as null and alternative hypotheses about the appropriate regression coefficients. Obtain the test statistic, state the distribution of the test statistic under the null hypothesis, and report the p-value for the test. State your conclusions. 9. Refer to the regression model obtained in Question 6 and compute any necessary anti-logarithms to answer the following questions. Suppose a migrant species has a small size and has, on average, 1.9 numbers of pairs. (a) What is the predicted value of extinction time for the species? (b) Obtain the 95% confidence interval for the mean extinction time of this species. In addition, obtain the 95% prediction interval for a species with these characteristics. Which of the two intervals is wider? Explain briefly. 10. Briefly summarize (2-3 sentences) the results of the statistical analysis. What is the scope of the inferences in the study? List some other variables that may affect the extinction time but were not considered in the study. 3 LAB ASSIGNMENT 4 MARKING SCHEMA Proper Header: 10 points Question 1 (a) Scatterplot of time versus number of pairs: 2 points Comment on pattern of association and implication: 3 points Difference in pattern by species size: 1 point Outlier: 2 points (b) Homogeneity of association: 2 points Question 2 Pearson's correlation coefficient: 2 points Comment on sign and magnitude of coefficient: 2 points Consistency: 1 point Question 3 (a) Scatterplot of ln(time) versus pairs: 2 points Comment on pattern of association and comparison: 2 points (b) Pearson's correlation coefficient: 2 points Comment on magnitude and comparison: 2 points Question 4 (a) Model definition: 2 points Assumptions: 2 points (b) Estimated regression line: 2 points (c) Significance of overall model: 6 points itemized as follows Hypotheses: 2 points Test statistic: 1 point Null distribution: 1 point P-value: 1 point Conclusion: 1 point (d) Percentage of variation: 2 points (e) Significance of each explanatory variable: 3 points (1 point each) Question 5 (a) Residual plot: 2 points Comment on pattern and validity of assumption: 2 points Outliers: 2 points (b) Normal probability plot: 2 points Comment on pattern and validity of assumption: 2 points (c) Influential observations and their case statistics: 4 points Question 6 (a) Explanation of the effect of cases 61 and 62: 2 points (b) Estimated regression line: 2 points (c) Percentage of variation explained: 2 points Comparison with untransformed scale: 1 point 4 (d) Residual plot: 2 points Comment on pattern and validity of assumption: 2 points (e) Normal probability plot: 2 points Comment on pattern and validity of assumption: 2 points Question 7 Hypothesis test: 9 points itemized as follows Hypotheses: 2 points Test statistic: 4 points Null distribution: 1 point P-value: 1 point Conclusion: 1 point Question 8 Model: 2 points Hypotheses: 2 points Test statistic: 1 point Null distribution: 1 point P-value: 1 point Conclusion: 1 point Question 9 (a) Predicted value: 2 points (b) 95% confidence interval: 2 points 95% prediction interval: 2 points Wider interval and explanation: 2 points Question 10 Summary of results: 2 points Scope of inference: 2 points Other variables: 2 points TOTAL = 110 Lab developed by Adeniyi Adewale Grant: G227150109 - Acc1Time Science Revised by Henryk Kolacz 5 $FL2@(#) IBM SPSS STATISTICS 64-bit MS Windows 21.0.0.0 #### ###########>#########Y@02 Jul 1412:17:09 ###########################SPECIES #################### #################### ######################TIME ################# #####PAIRS #####################SIZE #################### #### #################STATUS #################### ############################################################## ################################################################### ##################=###SPECIES=species TIME=time PAIRS=pairs SIZE=size STATUS=status########################>###################_###species: $@Role('0' )/time:$@Role('0' )/pairs:$@Role('0' )/size:$@Role('0' )/status:$@Role('0' )############ ###windows-1252#######eSparrowhawk ###p=#@L R fBuzzard ###"#@L R Kestrel ### Zd#@####)\\?L R Peregrine ###@`?######?L R Grey_partridge ###@3!@######@L R Quail e####T?L M Red-legged_partridge ####@#######@L R Pheasant####@#####@L R gWater_rail ###*0@L R Corncrake ###A#@#####@L M Moorhen ###`B @### X9#@L ieR Coot L R Lapwing ####2#@###@ #@L M Golden_plover e###Q?L M Ringed_plover ####;@###p=#@L R Curlew #####@#### #@L M hRedshank######@L M Snipe ### #!0@######@L M Stock_dove ###`;#@####)\\@L R Rock_dove ###`B@### @L R Wood_pigeon ####-2#@#######@L R Cuckoo ###@33#@###G?L M Short-eared_owl f####+#@L R Little_ofwl #######@L R nMagpie #######@L R Carrion_crow ####Y#@###Q#@L R Skylark ### #!@@###@z#@S R Swallow ###h#@#### #@S M House_martin ### G#@iS M Yellow_wagtail e######?S M Pied_wagtail ###j#@###(#@S R Meadow_pipit ###I #@###`ff#@S R Wren Dunnock ####-2#@###`ff#@S @S R Stonechat ####8&@###`ff!@S hR Robin ###I R #### #@###`#@S R Wheatear####T#@###Q#@S M Blackbird ####V# @#####@S R Song_thrush ###K7?###@33?S R Mistle_thrush ###Q?### G?S R eGrasshopper_warbler #####+?S M Sedge_warbler ###@z @###`ff?S M Whitethroat ####/#@#####@S M Willow_warbler ###? ######?eS M Chiffchaff eS M Goldcrest eeS R Spotted_flycatcheer ####?S M Great_tit ###v>#@#######@S R Blue_tit###`ff @######?S R fYellowhammer #######@S R Reed_bunting ####M#@#####@S R Chaffinch ####? ####@S R Goldfinch ####T?######?S R Redpoll eeS R Linnet ###rh#@#######@S R House_sparrow ###$##@#######@S R Tree_sparrow ###@5^?####)\\#@S R Starling###@`D@###p='@S eeR Pied_flycatcher S eeM Siskin S R Raven ###xiM@####@L R Jackdaw ##### #V#@###@z#@L R