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# HW2solns - Solutions Homework#2 Protein Consumption...

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Solutions - Homework #2 Protein Consumption Introduction Food consumption is of broad interest in the modern world. Patterns of food consumption are related to issue in overall health, obesity, nutrition, economics and the environment. This analysis is concerned with the relationship between meat consumption and consumption in other food groups in 25 European countries. It is of interest to determine how meat consumption is related to consumption in the other food groups. The data include measures of red meat, white meat, eggs, milk, fish, cereals, starchy foods, nuts (which includes pulses and oil-seeds), fruits & vegetables. The units of the variables are not reported, but are assumed to be of the form (unit weight)/(unit time). Methods A principal component factor analysis was performed to ascertain the relatedness of consumption across the different food groups. An preliminary factor analysis was performed to assess the cor- relation structure and the sampling adequacy. SAS reports the partial correlations, which were all small to moderate, indicating a structure conducive to factor analysis. The variables fish and fruits & vegetables have low Kaiser’s sampling adequacy: 0.3704 and 0.3838, respectively. While these measures of sampling adequacy are considered low and make the variables candidates for deletion, the variables were retained because of high interest in these food categories. We used the eigenvalue criterion for choosing the number of factors, cross checked by the proportion of variation explained by the factors. Both the quartimax and varimax rotation methods were performed, in order to choose the rotation that gave the highest interpretability. All analyses were performed in SAS. Variable Red Meat White Meat Eggs Milk Fish Cereals Starch Nuts Red Meat 1 -0.38485 0.4247 0.03148 -0.20075 -0.19904 -0.18009 -0.1564 White Meat -0.38485 1 0.4905 -0.35694 -0.43608 -0.17071 -0.06826 -0.56192 Eggs 0.4247 0.4905 1 0.30681 -0.1712 -0.38474 0.31877 0.34529 Milk 0.03148 -0.35694 0.30681 1 -0.11749 -0.15863 -0.22915 -0.50885 Fish -0.20075 -0.43608 -0.1712 -0.11749 1 -0.66229 0.26044 -0.01814 Cereals -0.19904 -0.17071 -0.38474 -0.15863 -0.66229 1 -0.01474 0.21437 Starch -0.18009 -0.06826 0.31877 -0.22915 0.26044 -0.01474 1 -0.30275 Nuts -0.1564 -0.56192 0.34529 -0.50885 -0.01814 0.21437 -0.30275 1 1

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Results and Conclusions Quartimax Rotation Variable Factor1 Factor2 Factor3 Red Meat 0.88819 -0.08046 -0.05557 Milk 0.80513 0.18256 0.07915 Eggs 0.70595 0.53234 0.07487 Cereals -0.66813 -0.36376 -0.53548 White Meat 0.22134 0.90478 -0.18394 Nuts -0.5074 -0.66631 -0.19927 Fish 0.13583 -0.18679 0.92462 Starch 0.11214 0.54789 0.65973 Varimax Rotation Variable Factor1 Factor2 Factor3 Red Meat 0.89314 0.02733 -0.00079 Milk 0.77107 0.27295 0.13702 Eggs 0.63093 0.60846 0.13808 White Meat 0.12144 0.93136 -0.13884 Nuts -0.41058 -0.71283 -0.25382 Fish 0.10153 -0.20718 0.92468 Starch 0.00551 0.52981 0.68358 Cereals -0.58608 -0.41786 -0.58938 The initial factor procedure identified 3 factors with eigenvalues greater than 1. These three fac- tors explained about 81% of the variation, with the first factor explaining about 50% and the next two explaining about 18% and 13% respec- tively. There were some cross loadings in the initial factor rotation. The quartimax and vari- max rotation loadings are shown below, with the factors loadings grouped for interpretation. The different rotation lead to different groupings and hence, possibly, different interpretations. Both
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