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241

extrapolating linear regression

DO NOT extrapolate beyond data, can't assume relationship continues to be linear

242

linear regression Ho

Slope is zero (β = 0), number of dees cannot be predicted from predator mass

243

linear regression Ha

slope is not zero (β ≠ 0), number of dees can be predicted from predator mass (2 sided)

244

Hypothesis testing of linear regression

testing about the slope:
–t-test approach
–ANOVA approac

245

Putting linear regression into words

Dee rate = 3.4 - 1.04(predator mass)
Number of dees decreases by about 1 pre kilo of predator mass increase

246

testing about the slope, t-test approach

test statistic t = b–β_o / SE_b
SE_b = √MSresidual/Σ(Xi-Xbar)^2
MSres. = Σ(Yi-Yhat)^2 / n-2
critical t = t_α(2),df
df = n - 2
compare statistic, critical

247

testing about the slope, ANOVA approach

source of variation: regression, residual, total
sum of squares, df, mean squares, F-ratio

248

calculating testing about the slope, ANOVA approach

SSregres = Σ(Yi^ - Ybar)^2
SSresid. = Σ(Yi-Yi^)^2
MSreg. = SSreg/df df=1
MSresid = SSres/df df=n-2
F-ratio = MSreg/MSres.
SStotal = Σ(Yi-Ybar)^2
df total = n-1

249

interpreting ANOVA approach to linear regression

If Ho is true, MSreg. = MSres

250

% of variation in Y explained by X

R^2 = SSreg/SStotal
a% of variation in Y can be predicted by X

251

Outliers, linear regression

create non-nomral Y-value distribution, violate assumption of equal variance in Y, strong effect on slope and intercept; try not to transform data

252

linear regression assumptions

linear relationship
normality of Y at each X
variance of Y same for every X
random sampling of Y's

253

detecting non-linearity

look at the scatter plot, look at residual plot

254

checking residuals

should be symmetric above/below zero
should be more points close line (0) than far
equal variance at all values of x

255

non-linear regression

when relationship is not linear, transformations don't work, many options- aim for simplicity

256

quadratic curves

Y = a + bX + cX^2
when c is negative, curve is humped
when c is positive, curve is u shaped

257

multiple explanatory variables

improve detection of treatment effects
investigate effects of ≥2 treatments + interactions
adjust for confounding variables when comparing ≥2 groups

258

GLM

general linear model; multiple explanatory variables can be included (even categorical); response variable (Y) = linear model + error

259

least-squares regression GLM

Y = a + bX
error = residuals

260

single-factor ANOVA GLM

Y = µ + A
error = variability within groups
µ = grand mean

261

GLM hypotheses

Ho: response = constant; response is same among treatments
Ha: response = constant + explanatory variable

262

constant

constant = intercept or grand mean

263

variable

variable = variable x coefficient

264

ANOVA results, GLM

source of variation: Companion, Residual, Total
SS, df, MS, F, P

265

ANOVA, GLM F-ratio

MScomp. / MSres.

266

ANOVA, GLM R^2

R^2 = SScom. / SStot.
% of variation that is explained

267

ANOVA, GLM, reject Ho

Model with treatment variable fits the data better than the null model but only 25% of the variation is explained

268

Multiple explanatory variables, goals

improve detection of treatment effects
adjust for effects of confounding variables
investigate multiple variables and their interaction

269

design feature for improving detection of treatment effects

blocking

270

design feature for adjusting for effects of confounding variables

covariates