Introduction to Biostatistics
Introduction to Biostatistics STAT 541
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Mrs. Triston Collier
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This 2 page Class Notes was uploaded by Mrs. Triston Collier on Thursday September 17, 2015. The Class Notes belongs to STAT 541 at University of Wisconsin - Madison taught by Staff in Fall. Since its upload, it has received 43 views. For similar materials see /class/205085/stat-541-university-of-wisconsin-madison in Statistics at University of Wisconsin - Madison.
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Date Created: 09/17/15
lsm or Fischer 8152008 Appendix A4 Regression Models Exponential Growth1 Exponential Growth Consider a somewhat idealized example of how to use a logarithm transformation on exponential growth data Assume we start with an initial population of 100 cells in culture and they grow under ideal conditions exactly doubling their numbers once every hour Let X time hours Y population size suppose we obtain the following data X 0 1 2 3 4 Y 100 200 400 800 1600 A scatterplot reveals typical exponential aka geometric growth see Figure 1 A linear t of these data points X Y will not be a particularly good model for it but there is nothing to prevent us either statistically or mathematically from proceeding this way Their least squares regression line also shown in Figure 1 is given by the equation Y 100 360 X with a coef cient of determination r2 0871 Exercise Verify these claims Although I2 is fairly close to 1 there is nothing scienti cally compelling about this model there is certainly nothing natural or enlightening about the regression coef cients 100 and 360 in the context of this particular application This illustrates the drawback of relying on r2 as the sole indicator of the t of the linear model One alternative approach is to take the logarithm we will use common logarithm base 10 ofthe response variable Y7 which is possible to do since Ytakes positive values 7 in an attempt to put the population size on the same scale as the time variable X This gives log10Y 20 23 26 29 32 Notice that the transformed response variable increases with a constant slope 03 for every onehour increase in time the hallmark of linear behavior Therefore since the points X log10Y are collinear their least squares regression line is given by the equation log10Y 2 03 X Verify this by computing the regression coef cients Given this we can now solve for the population size directly Inverting the logarithm I 10203X 10 2 X 10 0393X via a law of exponents This exponential growth model is a much better t to the data see Figure 2 In fact it s exact check it for X 0 1 2 3 4 and makes intuitively reasonable sense for this application The population size Y at any time X is equal to the initial population size of 100 times 2 raised to the X power since it doubles in size every hour This is an example of unrestricted exponential growth The technique of logistic regression applies to restricted exponential growth models among other things IsmanxschEr 8152008 Appmd x m4 Regessmn Mndels Expnnenual Gmwvhrl Fxgurel C c a 8 a 5 g a a 0 1 2 3 4 x FxgureZ y 0 1 2 3 4 X
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