We are 95 confident that the interval covers the true

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We are 95% confident that the interval covers the true β If the interval does not cover 0, we are 95% confident the true value of β is not zero “^” (hat) stands for estimated parameter ^ ^
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Example Staessen et al. (1992) investigated the relationship between lead exposure and kidney function. Background : Heavy lead exposure can lead to kidney damage Kidney function decreases with age People accumulate small amounts of lead as they get older Researchers wanted to know whether accumulation of lead could explain some of the age-related decrease in kidney function. Researchers collected data on lead exposure, creatinine clearance (measure of kidney function), age, and other covariates on 965 men 38 Here, age is a confounder
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Example (cont’d) 39 Variable Meaning Units X 1 Log 10 (serum lead) Serum led was in ࠵? g/L X 2 Age Years X 3 Body mass index kg/m 2 X 4 Log(GGT) Serum GGT was in U/L X 5 Diuretics 1 = yes (took diuretics) 0 = no (never took diuretics) Y Creatinine clearance ml per minute Model : Y i = α + β 1 X i1 + β 2 X i2 + β 3 X i3 + β 4 X i4 + β 5 X i5 + ࠵? i Used log(serum lead) rather than serum lead, because they expected the effect of lead to be multiplicative, i.e., a doubling of lead concentration would lead to an equal effect on creatinine clearance, no matter the starting value.
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Example (cont’d) 40 Question: after adjusting for effects of the other variables, is there a linear relationship between the logarithm of lead concentration and creatinine clearance? Result: Best-fit value of β 1 was -9.5 ml/min (95% CI: -18.1 to -0.9 ml/min) I.e., after accounting for differences in other variables, an increase in log(lead) of 1 unit is associated with a decrease in creatinine clearance of 9.5 ml/min (95% CI: -18.1 to -0.9 ml/min) Note: 1 unit increase in log(lead) means 10-fold increase in lead concentration (since log base 10 was used) Since the 95% CI does not include 0, we know that P<0.05.
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R 2 : How well does the model fit the data? 41 R 2 (coefficient of multiple determination) - measures the proportion of variability in the response variable that is explained by the model (i.e., set of variables). In the example, R 2 is 0.27. This means that 27% of variation in creatinine clearance is explained by the model. The remaining 73% is explained by other variables (not included in the study) and random variation Note: can calculate partial R 2 for each variable that quantifies the proportion of variation in the response explained by each predictor after accounting for other predictors
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Intuitive Biostatistics Harvey Motulsky Copyright © 2018 Oxford University Press Figure 37.2. The meaning of R 2 in multiple regression. R 2 : How well does the model fit the data? Scatter plot of measured creatinine clearance vs creatinine clearance predicted by the model. R 2 for the predicted and actual values is identical the R 2 from the model.
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R 2 vs adjusted R 2 43 R 2 is commonly used as a measure of fit in multiple regression models.
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