Lecture19 - Lecture 19: Regression Analysis II Confidence...

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Lecture 19: Regression Analysis II Confidence Intervals, Prediction Intervals Sources of Information Motulsky: Chapter 18, 19 Triola et al.: Chapter 9 Sokal & Rohlf: Chapter 14
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Predicting values of the dependent variable We can use a regression line (estimated from data obtained from a sample) to predict the mean values for the dependent variable (Y) based on values of the independent variable (X). Y = a + b X Example: mean insulin sensitivity = -486.542 + (37.2077 * %C20-22) For a man with %C20-22 of 20, the predicted average insulin sensitivity = -486.542 + (37.2077*20) = 257.612 mg/m 2 /min
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Confidence in the Predictions Do you have confidence in this prediction? Of course not. First, we know that if we take another sample of 13 men, we would get slightly different values of the slope ( b ) and intercept ( a ). Also, we suspect that not all men with %C20-22 have exactly the same insulin sensitivity. So, all kinds of values are possible.
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Confidence in the Predictions To make your prediction more useful, you must estimate the variability associated with it. For example, if you know that for a man with %C20-22=20, the range of plausible values of average insulin sensitivity is from –100 to 500 mg/m 2 /min, then you know that your prediction is pretty useless. However, if the range of values is from 255 to 265 mg/m 2 /min, your estimate is more useful.
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The interpretation should be familiar. • The best-fit line determined from a particular sample of subjects is unlikely to really be the best-fit line for the entire population . • If the assumptions of linear regression are true, you can be 95% sure that the overall best-fit regression line (for the population) lies somewhere within the space enclosed by the Confidence Interval. (Motulsky Fig. 19.1) This figure shows the 95% confidence interval for the regression line.
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Really, it means that if we repeated the experiment over and over, in the long-run 95% of the similarly constructed CI’s would include the true best-fit line for the population. (Motulsky Fig. 19.2) (Motulsky Fig. 19.1)
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Before you can estimate the variability of your predicted value, you must decide which of two types of predictions (or intervals) you are interested in making: 1. Do you want to predict the average insulin sensitivity for all healthy adult men who have a %C20-22 = 20? [= confidence interval ] Or, 2. Do you want to predict the insulin sensitivity for a particular man; say, the next man you randomly sample who has a %C20-22 = 20 with a certain level of confidence? [= prediction interval ] In both cases, the predicted value is the same. ( from Y = a + bX ) What differs is the variability . We know that means vary less than individual observations. So, we shouldn’t be surprised that you can predict the average insulin
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This note was uploaded on 07/10/2010 for the course BIOL 361 taught by Professor Hall during the Winter '08 term at Waterloo.

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Lecture19 - Lecture 19: Regression Analysis II Confidence...

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