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23 Pages

### 17LinearCombos

Course: EXST 7087, Fall 2009
School: LSU
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Word Count: 1062

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Techniques Statistical I EXST7005 Linear Combinations Linear combinations This is a function of random variables of the form aiYi where ai is a constant and Yi is a variable. s Generic Example: We want to create a score we can use to evaluate students applying to LSU as freshmen. s Score=a(VerbalSAT)+b(MathSAT)+c(GPA) s where a, b and c are the constants and s VerbalSAT, MathSAT and GPA are the variables (they...

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Techniques Statistical I EXST7005 Linear Combinations Linear combinations This is a function of random variables of the form aiYi where ai is a constant and Yi is a variable. s Generic Example: We want to create a score we can use to evaluate students applying to LSU as freshmen. s Score=a(VerbalSAT)+b(MathSAT)+c(GPA) s where a, b and c are the constants and s VerbalSAT, MathSAT and GPA are the variables (they vary among students). s Linear combinations (continued) s Score=a(VerbalSAT)+b(MathSAT)+c(GPA) we need to choose values of a, b and c if a=1/3 and b=1/3 and c=1/3 then we have an average (a+b+c)/3. If a=1 and b=1 and c=1 we have a simple sum (a+b+c). But VerbalSAT and MathSAT are values in the hundreds and GPA is around 2 or 3. So we might choose a = b = 1/100 and c = 1. Any of these is a linear combination. Mean and Variance s So what is the mean value of our linear combination, and can we put a variance on i (to get a confidence interval)? Linear combination: aiYi Expected value: ai Yi Estimate of mean: aiYi Mean and Variance (continued) s Variance of the linear combination. The variance of a linear combination is the sum of the variances of the individual variances (wi squared coefficients) plus twice the covariance of the variables (with both coefficients). Estimate of the variance: a2iSi2 + 2aiaj(Covariances) Mean and Variance (continued) s For example, with the Score we calculated previously, the variance might be VAR(Score) = a2VAR(Verbal) + b2VAR(Math) + c2VAR(GPA) + (2abCOV(Verbal,Math) + 2ac(Verbal,GPA) + 2bcCOV(Math,GPA) Mean and Variance (continued) HOWEVER, if the variables are independent the covariance can assumed to be zero. s The linear combination reduces to the sum o the variances of the individual variables (wit squared coefficients. s VAR(Score) = a2VAR(Verbal) + b2VAR(Math) + c2VAR(GPA) Utility of linear combinations As the course progresses we will see applications of linear combinations to almos everything. s Two sample t-test: H0: 1 - 2 = 0, estimate by Y1 -Y2 = 0. This is a linear combination Suppose you want to test 0.5*Y1- 2*Y2 = 0 s Regression: The model, Yi = b0 + b1X + ei is linear combination. s Utility of linear combinations (continued) s Analysis of variance: We will look at contrasts to test for differences between means similar to a two sample t-test (but wit more means). For example, test the hypothesis that H0: 2 1 - 2 - 3 = 0 An application: Stratified Random Sampling s Suppose we want to estimated the number o ducks in an area on the Louisiana coast. Th area of interest is 300 acres. We fly 9 transects, counting ducks for 1/4 mile on either side of the plane, and from each transect we estimate the number of ducks per acre. We can then calculate an estimate and a confidence interval for that estimate. An application: Stratified Random Sampling (continued) s Sampling results and calculations Sample value 8 19 30 23 56 89 2024 1732 1122 Statistic Value n= 9 sum = 5103 mean = 567 var = 684349.25 std dev = 827.25 std error = 91.917 t-value = 2.306 acres = 300 An application: Stratified Random Sampling (continued) s Estimates and confidence interval Sample value 8 19 30 23 56 89 2024 1732 1122 Statistic mean acres est total variance std error half width CL-lower CL-upper Value 567 300 170100 61591432500 82725.40 91.917 190765 -20665 360865 An application: Stratified Random Sampling (continued) Where the true number of ducks is can DUCKS we state our results as a probability statement. s P(-20665 Total Ducks 360865)=0.95 s Note that this calculation is for the total number of ducks on the 300 acres of interes s Stratification Now lets suppose we noticed that the ducks species of interest were primarily a fresh water species, and occurred only infrequently in the brackish and saline areas of the 300 acres of interest. s The 3 habitat types would be called "strata", and we could estimate the number for each strata separately (and INDEPENDENTLY). s Stratification (continued) s Results for separate areas Obs 1 2 3 n= sum = mean = var = std dev = std error = Saline 8 19 30 3 57 19 121 11 3.667 Brackish 23 56 89 3 168 56 1089 33 11 Fresh 2024 1732 1122 3 4878 1626 211828 460.248 153.416 Stratification (continued) s Linear combination of independent means. t-value acres est total variance std error half width CL-lower CL-upper 2.306 300 170100 2130380000 15385.347 35478.70 134621.30 205578.70 Stratification (continued) Old and new estimates. s P(-20665 Total Ducks 360865)=0.95 s P(134621 Total Ducks 205579)=0.95 s Why does stratification give a smaller interv width? Because we replaced one sample with a very large variance, 61591432500, wit 3 samples with smaller variances: 121, 1089 and 211828. s Stratification (continued) s You may recall that one way of increasing power is to reduce the variance. We did not dwell on this previously because the only mechanism I could suggest at the time for reducing variance was "improving measurement error". Now we have another method of reducing variance, stratification. This involves sampling smaller homogeneous units instead of one large heterogeneous unit. Stratification (continued) s Is the fact that the 3 variances were not similar a problem? No, nowhere in working with linear combinations did we state that th variances had to be similar. Later we will fin that this can be advantageous, but it is not necessary for this type of analysis. Summary The use of "Linear Combinations" is a rathe generic technique with many applications throughout statistics. We will see them aga in the two sample t-test, regression, and ANOVA. Sampling is another example of th application. s It is important to determine if the variables i the linear combination are independent or not. If they are, the covariances can be considered to be zero. s Summary (continued) s The linear combination and it's variance is calculated as Linear combination = aiYi Variance = a2iSi2 + 2aiaj(Covariances) Where if we can assume the variables are independent, 2aiaj(Covariances)=0. This will b very important later. We will assume independence in t-test and ANOVA, and parts o regression, but not all of regression! A final note on linear combinations We have been assuming "independence" fo a while. We sample at random to obtain independence, and to get a good representative sample. s But do "linear combinations" have anything to do with the simpler calculations we talked about earlier when we assumed independence, say some hypothesis test of the mean? s A final note on linear combinations (continued) I'm glad you asked. As a matter of fact the mean is calculated as s mean = (Y1 + Y2 + Y3 + ... + Yn) / n s and that is a linear combination. Fortunatel for us we do not have to consider the COVARIANCES of individual observations because they are sampled INDEPENDENTLY s
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