Bio286 Worksheet 4: Multiple Regression, Non-Linear Regression - Answer Key
1bi: The null hypothesis for these relationships is Ho: slope of y on x =0. Note that this is the same as
Ho: slope of y on x = 𝑦̅.
1biv-vi:
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Bio286 Worksheet 4: Multiple Regression, Non-Linear Regression - Answer Key
1bvii and 1bviii: Always use adjusted r2
Independent Variable
Slope
P-Value
r2 for model
Conclusion with
respect to Ho
Diatom abundance
1.05
<0.0001
.968
Reject
Other limpets
-1.50
0.365
0 (r2 cannot be
negative)
Accept
Tide Height
8.03
0.0127
.27
Reject
Predators
4.48
0.58
0
Accept
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Bio286 Worksheet 4: Multiple Regression, Non-Linear Regression - Answer Key
Hence you would reject the hypotheses that predators and other limpets are associated with the Focal
Limpet density and would accept the hypotheses that food and tide height are associated with the
density of Focal limpets. Note that the sign of the slope will always be the same as the sign of the tvalue.
1cv:
The VIF scores indicate that there is no co-linearity among variables that affects the results of the
analysis. The Estimate values represent the model intercept and the slopes of the relationship between
the X variables (eg diatom abundance) after accounting for all the X variables. Here there are three
terms that are significant (diatom abundance (+), other limpets (-) and tide height (+). All are significant
at the p<0.0001 level. In the simple regression analyses tide height was significant but much less so and
other limpets was not significant. The density of predators is not significant in the multiple regression or
the simple regression analyses. The r2 score for the multiple regression analysis is higher than any of the
values for the simple regressions.
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Bio286 Worksheet 4: Multiple Regression, Non-Linear Regression - Answer Key
1cviii
1cix:
A multiple regression yields a linear equation in the slope intercept form:
Focal-limpet = -23.39+1.01(diatom abundance)-1.05(other limpets)+.98(tide height)-0.07 (predators).
Given that the density of predators was not significant in the model you could drop the predator effect.
The answer is around 110- 111 depending on rounding.
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Bio286 Worksheet 4: Multiple Regression, Non-Linear Regression - Answer Key
2b. use the loess smoother (curved line icon) and play with Lambda to the desired tension.
2c and d.
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Bio286 Worksheet 4: Multiple Regression, Non-Linear Regression - Answer Key
Model:
AIC corrected
Mean (1 parameter): 𝑦 = 𝑦̅
447.81
Linear Regression (2 parameter): 𝑦 = 𝑏0 + 𝑏1 + 𝜀
414.75
Gompertz 3 parameter: 𝑦 = 𝑎𝑒 −𝑒
𝑛𝑎𝑡𝑢𝑟𝑎𝑙 𝑒𝑥𝑝𝑜𝑛𝑒𝑛𝑡
−𝑏(𝑥−𝑐)
, where 𝑒 =
Biexponential 5 parameter: 𝑦 = 𝑎 + 𝑏𝑒 −𝑐𝑥 + 𝑑𝑒 −𝑓𝑥 , where 𝑒 =
𝑛𝑎𝑡𝑢𝑟𝑎𝑙 𝑒𝑥𝑝𝑜𝑛𝑒𝑛𝑡
348.12
346.81
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