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# 511_Chap8 - A Closer Look at Assumptions for Simple Linear...

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Stat 511 Chap 8 1 A Closer Look at Assumptions for Simple Linear Regression Chapter 8

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Stat 511 Chap 8 2 Case Study 1: Island Area and Number of Species observational study on an important issue in biogeography and conservation biology – the relationship between island size and number of species study carried out on islands in West Indies, involves reptile and amphibian species ( herps ) based on immigration and extinction rates relative to island area, theory suggests the relationship S=CA γ taking logs of both variables, log(S) = log(C) + γ log(A) intercept slope
Stat 511 Chap 8 3 0 20 40 60 80 100 120 SPECIES 0 10000 20000 30000 40000 AREA 1.5 2 2.5 3 3.5 4 4.5 5 logspecies -2 0 2 4 6 8 10 12 logarea

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Stat 511 Chap 8 4 Case Study 2: Breakdown Times designed experiment for different voltages, batches of insulating fluid were subjected to constant current time until the insulating property of the fluid broke down was observed log scale used for breakdown times several replicate runs for each voltage data to be used for predicting breakdown times
Stat 511 Chap 8 5 -3 -2 -1 0 1 2 3 4 5 6 7 8 logtime 24 26 28 30 32 34 36 38 40 VOLTAGE

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Stat 511 Chap 8 6 ASSUMPTIONS x y EQUAL STANDARD DEVIATION The standard deviation( σ ) of Y is the same for all values of X. INDEPENDENCE Observations are independent of each other. μ y = β 0 + β 1 x LINEARITY The mean response, μ , has a straight-line relationship with X, ( μ {Y|X}= β 0 + β 1 X). Simple Linear Regression Model NORMALITY For any given X, Y is normally distributed around μ {Y|X}.
Stat 511 Chap 8 7 Robustness to Assumptions Linearity – no (not very robust) all inferences (tests and confidence intervals for coefficients, means and predictions) are biased if means do not follow a straight line model

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Stat 511 Chap 8 8 Robustness to Assumptions Normality – yes and no inferences about coefficients and means are fairly robust to non-normality only concerned about non-normality when distributions are long-tailed and sample sizes are small to moderate predictions are not robust to non-normality
Stat 511 Chap 8 9 Robustness to Assumptions Equality of St. Dev. s – yes and no inferences about coefficients and means are somewhat robust to non-equality of standard deviations (similar considerations apply as for one-way ANOVA) estimates are always unbiased, but standard errors may be inaccurate predictions are not robust to non-equality of standard deviations

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Stat 511 Chap 8 10 Robustness to Assumptions Independence – no (well yes in one sense) estimates of coefficients and means are actually unbiased if there are cluster or serial effects but all standard errors are seriously affected if there are cluster or serial effects
Stat 511 Chap 8 11 Graphical Tools for Model Assessment 1. Scatterplot of Y vs. X 2. Scatterplot of Residuals vs. Fitted Values 3. Normal Probability Plot of Residuals 4. Scatterplot of Residuals vs. Time Order (row) easily produced in JMP available in JMP {

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