Flashcards in BIO 330 Deck (379):

1

## sampling error imposes

###
imprecision (accuracy intact)

caused by chance

2

## sampling bias imposes

### inaccuracy (precision intact)

3

## accurate sample

### unbiased

4

## precise sample

### low sampling error

5

## good sample

###
accurate

precise

random

large

6

## 2 types of data

###
numerical

categorical

7

## numerical data

###
continuous

discrete

8

## categorical data

###
nominal

ordinal

9

## types of variable

###
response

explanatory

10

## response variable

###
dependent

outcome

Y

11

## explanatory variable

###
independent

predictor

x

12

## subsamples treated as true replicate

### pseudoreplication

13

## subsamples are useful for

### increasing precision of estimate for individual samples (multiple samples from same site averaged)

14

## contingency table

###
explanatory- columns

response- rows

totals of columns and rows

15

## 2 data descriptions

###
central tendency

width

16

## central tendency

###
mean

median

mode

17

## width (spread)

###
range

standard deviation

variance

coefficient of variation

IQR

18

## effect of outliers on mean

###
shifts mean towards outliers- sensitive to extremes

median doesn't shift

19

## sample variance s^2 =

### sum( Y_i - Ybar )^2 / n-1

20

## coefficient of variation CV =

### 100% ( s / Ybar )

21

## high CV

### more variability

22

## skewed box plot

###
left skewed- more data in 'bottom'- first quartile

right skewed- more data in 'top'- 3rd quartile

23

## when/why random sample

###
uniform study area

removes bias in sample selection

24

## when/why systematic sample

### detect patterns along gradient- fixed intervals along transect/belt

25

## using quadrats

###
more better

stop when mean/variance stabilize (asymptote)

26

## what does changing n do to sampling distribution

### reduces spread (narrows graph) - increases preciesion

27

## standard error of estimate SE_Ybar =

### s / sqr rt (n)

28

## SD vs. SE

###
SD- spread of distribution/deviation from mean

SE- precisions of an estimate (ex. mean)

29

## 95% CI ~=

### +/- 2SE

30

## kurtosis

###
leptokurtic- sharper peak (+)

platykurtic- rounder peak (-)

mesokurtic- normal (0)

31

## Normal distribution, 1SD

### ~2/3 of the area under the curve (2SD = 95%)

32

## random trial

### process/experiment with ≥2 possible outcomes who occurrence can not be predicted

33

## sample space

### all possible outcomes

34

## event

### any subset of the sample space (≥1 outcome)

35

## mutually exclusive events

### P[A and B] = 0

36

## mutually exclusive addition rule

### P[7U11] = P[7} + P[11]

37

## general addition rule

### P[AUB] = P[A] + P[B] - P[A and B]

38

## multiplication rule

###
independent events

P[A and B] = P[A] x P[B]

39

## conditional probability

### P[A I B] = P[A and B] / P[B]

40

## collection of individual easily available to researcher

### sample of convenience

41

## random sample

### ever unit has equal opportunity, selection of unit independent, minimizes bias, possible to measure sampling error

42

## problem with sample of convenience

### assume unbiased/independent- no guarantee

43

## volunteer bias

### health conscious, low income, ill, more time, angry, less prudish

44

## frequency distribution

### describes # of times each value of a variable occurs in sample

45

## probability distribution

### distribution of variable in whole population

46

## absolute frequency

### # of times value is observed

47

## relative frequency

### proportion of individuals which have that value

48

## experimental studies can

###
determine cause and effect

*cause

49

## observational studies can

###
only point to cause

*correlations

50

## quantifying precision

### smaller range of values (spread)

51

## determining accuracy

### usually can't- don't know true value

52

## nominal categorical data with 2 choices

### binomial

53

## why aim for numerical data

### it can be converted to categorical if need be

54

## species richness

### discrete (count)

55

## rates

### continuous

56

## large sample

###
less effected by chance

lower sampling error

lower bias

57

## rounding

### round to one decimal place more than measurement (in calculations)

58

## higher CV

### more variability

59

## proportions

### p^ = # of observations in category of interest/ total # of observations in all categories

60

## sum of squares

###
it is squared so that each value is +, so they don't cancel each other out

n-1 to account for population bias

61

## CV used for

### relative measures- comparing data sets

62

## sampling distribution

### probability distribution of all values for an estimate that we might obtain when we sample a population, centred at true µ

63

## values outside of CI

### implausible

64

## how many quadrats to use

### till cumulative number of observations asymptotes

65

## law of total probability

###
P[A] = Σ P[B].P[A I B]

for all B_i 's

66

## null distribution

### sampling distribution for test statistic, if repeated trials many time and graphed test statistics for H_o

67

## Type I error

### P[Reject Ho I Ho true] = alpha

68

## reject null

### P-vale < alpha

69

## Type II error

### P[do not reject Ho I Ho false]

70

## Power

###
P[Reject Ho I Ho false]

increases with large n

decreases P[Type II E]

71

## test statistic

### used to evaluate whether data are reasonably expected under Ho

72

## p-value

### probability of getting data as extreme or more, given Ho is true

73

## statistically significant

###
data differ from H_o

not necessarily important- depends on magnitude of difference and n

74

## why not reduce alpha

### would decrease P[Type I] but increase P[Type II]

75

##
continuous probability

P[Y = y] =

### 0

76

## sampling without replacement

###
ex. drawing cards

(1/52).(1/51).(1/50)

77

## Bayes Theorem

### P[A I B] = ΣP[B I A].P[A] / P[B]

78

## P-value > alpha

###
do not reject Ho

data are consistent with Ho

79

## meaning of 'z' in standardization

### how many sd's Y is from µ

80

## standardization for sample mean, t =

### Ybar - µ / (s / sq.rt. n)

81

## CI on µ

###
Ybar ± SE.tcrit

SE of Ybar

t of alpha(1 or 2), degrees of freedom

82

## 1 sample t-test

### compares sample mean from normal pop. to population µ proposed by Ho

83

## why n-1 account for sampling error

### last value is not free to vary if mean is a specified value

84

## 1 sample t-test assumptions

###
data are a random sample

variable is normally distributed in pop.

85

## paired t-test assumptions

###
pairs are a random sample from pop.

paired differences are normally distributed in the pop.

86

## how to tell whether to reject with t-test

### if test statistic is further into tails than critical t then reject

87

## 2 sample design compares

### treatment vs. control

88

## 2 sample t-test assumptions

###
both samples are random samples

variable is normally distributed in each group

standard deviation in two groups ± equal

89

## degrees of freedom

###
1 sample t-test: n - 1

paired t-test: n - 1

2 sample t-test: n1 + n2 - 2

90

## confounding variables

###
mask/distort causal relationships btw measured variables

problem w/ observational studies

impossible to differentiate 1 variable

91

## experimental artifacts

###
bias resulting from experiment, unnatural conditions

problem w/ experimental studies

should try to mimic natural environment

92

## minimum study design requirements

###
knowledge of initial/natural conditions via preliminary data to ID hypotheses and confounding variables

controls to reduce bias

replication to reduce sampling error

93

## study design process

###
develop clear statement of research question

list possible outcomes

develop experimental plan

check for design problems

94

## developing a clear statement of research question

###
ID question, Ho, Ha

choose factors, response variable

what is being testes? will the experiment actually test this?

95

## list possible outcome of experiment

###
ID sample space

explain how each outcome supports/refutes Ho

consider external risk factors

96

##
develop experimental plan

based on step 1

###
outline different experimental designs

check literature for existing/accepted designs

97

## develop experimental plan based on step 2

###
what kind of data will you have- aim for numerical

what type of statistical test will you use

98

## minimize bias in experimental plan

###
control group

randomization

blinding

99

## minimize sampling error in experimental plan

###
replication

balance

blocking

100

## types of controls

###
positive

negative

101

## positive control

###
treatment that should produce obvious, strong effect

ensuring experiment design doesn't block effect

102

## negative control

### subjects go through all same steps but do not receive treatment- no effect

103

## maintaining power with controls

### add controls w/o reducing sample size- too many controls samples using up resources will reduce power

104

## placebo effect

### improvement in condition from psychological effect

105

## randomization

### breaks correlation btw explanatory variable and confounding variables (averages effects of confounding variables)

106

## blinding

###
conceals from subjects/researchers which treatment was received

prevent conscious/unconscious changes in behaviour

single blind or double blind

107

## better chance of IDing treatment effect if

### sample error/noise is minimized

108

## replication =

### smaller SE, tighter CI

109

## spacial autocorrelation

###
each sample is correlated w/ sample area

not independent (unless testing differences in that population)

110

## temporal autocorrelation

### measurement at one pt in time is directly correlated w/ the one before/after it

111

## balance =

### small SE, narrow CI

112

## blocking

###
accounts for extraneous variation by putting experimental units that are similar into 'blocks'

only concerned w/ differences within block- differences btw blocks don't matter

lowers noise

113

## factorial design

###
most powerful study design

study multiple treatments and their interactions

equal replication of all combinations of treatment

114

## checking for pseudoreplication

###
check degrees of freedom, very large- problem

overestimate = easier to reject Ho- pretending we have more power than we do

115

## determining sample size, plan for

### precision, power, data loss

116

## determining sample size, wanting precision

###
want low CI

n ~ 8(sigma/uncertainty)^2

uncertainty is 1/2 CI

117

## determining sample size, wanting power

###
detecting effect/difference

plan for probability of rejecting a false Ho

n~16(sigma/D)^2

D is min. effect size you want to detect

power is 0.8

118

## ethics

###
avoid trivial experiment

collaborate to streamline efforts

substitute models for live animals when possible

keep encounters brief to reduce stress

119

## most important in experimental study design

###
check common design problems

sample size (precision,power,data loss)

get a second opinion

120

## most important in observational study design

### keep track of confounding variables

121

## good skewness range for normality

### [-1,1]

122

## normal quantile plot

###
QQ plot

compares data w/ standardized value, should follow a straight line

123

## right skew in QQ plot

### above line (more positive data)

124

## Shapiro-Wilk test

###
works like Hypothesis test, Ho: data normal

estimate pop mean and SD using sample data, tests match to normal distribution with same mean and SD

p-value < alpha, reject Ho (don't want to reject)

125

## testing normality

###
Histogram

QQ plot

Shapiro-Wilk

126

## normality tests sensitive

###
especially to outliers, over-rejection rate

sensitive to sample size

large n = more power

127

## testing equal variances

### Levene's test

128

## Levene's test

###
Ho: sigma1 = sigma2

difference btw each data point and mean, test difference btw groups in the means of these differences

p-value < alpha reject (don't want to reject)

129

## how to handle violations of test assumptions

###
ignore it

transform data

use nonparametric test

use permutation test

130

## when to ignore normality

###
CLT- n >30 ----means are ~normally distributed

depends on data set though

can't ignore normality and compare one set skewed left with one skewed right

131

## when to ignore equal variances

###
n large, n1 ~ n2

3 fold difference in SD usually ok

132

## if can't ignore violation of equal variances

### Welch's t-test- computes SE and df differently

133

## most common transformations

###
log, arcsine, square-root

log- only in data all > 0

134

## nonparametrics

###
assume less about underlying distributions

usually based on rank data

Ho: ranks are same btw groups

sign test (instead of t test)

135

## sign test

###
compares median to median in Ho

each data pt- record whether above (+) or below (-) the Ho median

136

## if Ho is true in sign test

### half data will be above Ho, half will be below

137

## sign test p-value

### use binomial distribution-- probability of getting your measurement if Ho true, compare to alpha

138

## binomial

### P[Y≤y] = Σ(n choose y)(p)^y(1-p)^n-y

139

## Mann-Whitney U-test

###
compare 2 groups using ranks

doesn't assume normality

assumes distributions are same shape

rank all data from both groups together, sum ranks for individual groups

140

## Mann-Whitney U-test equation

###
U1 = n1n2 + [(n1(n1+1)/2] - R1

U2 = n1n2 - U1

141

## interpreting Mann-Whitney U-test

###
choose larger of U1, U2 (test statistics)- compare to critical U from U distribution (table E)

note that Ucrit = U_alpha,(2 sided), n1, n2

used n1, n2 not DF

U < Ucrit d.n.r. Ho (2 groups not statistically different)

142

## why Mann-Whitney doesn't use DF

### not looking at estimating mean/variance, just comparing the shapes

143

## problem with non-parametrics

###
low power- P[Type II] higher-- especially with low n

ranking data = major info loss

avoid use

Type I not altered

144

## comparing > 2 groups

###
ANOVA - analysis of variance

Ho: µ1 = µ2 = µ3 = µ4....

145

## why use ANOVA

### multiple t-tests to compare >2 groups increase Type I error- more tests = higher chance of falling within alpha

146

## P[Type I]

###
1 - ( 1 - alpha ) ^N

N is number of t-tests you do

ex. 5 groups- 10 unique tests- P[TI] = 0.4

147

## ANOVA tests

###
is there more variation btw groups than can be attributed to chance- breaks it down into: total variation, btw group variation, within group variation

maintains P[TI] = alpha

148

## between-group variation

### effect of interest (signal)

149

## within-group variation

### sampling error (noise)

150

## 2x2 ANOVA design

###
take 2 different variables-- look at all combinations and see if any effects between them in all directions

2 variables w/controls = 8 options

151

## Hypothesis test steps

###
State Ho, Ha

calculate test statistic

determine critical value of null distribution (or P-value)

compare tests statistic to critical value (or P-value to sig. level)

evaluate Ho using alpha

152

## why use alpha = 0.05

### balances Type I error and Type II error

153

## why are Type I and II errors conceptual

### we don't know whether or not Ho is actually true

154

## paired t-test is a type of

### blocking

155

## where does pseudoreplication happen/become a problem

### data analysis stage, doesn't happen at data collection stage (subsamples)

156

## ANOVA maintains

### P[Type I Error] = alpha

157

## ANOVA, Y bar

### grand mean, main horizontal line, test for differences between grand mean and group means

158

## ANOVA, Ho: F-ratio =

### ~1

159

## ANOVA, if Ho is true, MSerror

### = MS groups; same variation within and btw goups

160

## ANOVA, MSgroup > MSerror

### more variation between groups than within

161

## ANOVA, test statistic

###
F-distribution, F_0.05,(1),MSgroup DF, MSerror DF = critical value

compare critical value to F-ratio

this is a one sided distribution we are looking for whether F-ratio is bigger than critical value (strictly)

162

## ANOVA, F-ratio > F-critical

### Reject Ho.. at least one group mean is different than the others

163

## ANOVA, quantifying variation resulting from "treatment effect"

###
R^2 = SSgroups/SStotal

R^2 [0,1]

164

## ANOVA, high R^2

### more of the variation can be explained by the treatment, usually want at least 0.5

165

## ANOVA, R^2 = 0.43

### 43% of total variation is explained by differences in treatment

166

## ANOVA, R^2 = low values

### noisy data

167

## ANOVA assumptions

###
Random samples from populations

Variable is normally distributed in each k population

Equal variance in all k populations

168

## ANOVA unmet assumptions

###
large n, similar variances-- ignore

variances very different-- transform

non-parametric-- Kruskal-Wallis

169

## ANOVA, which group(s) were different

### Planned or Unplanned comparison of means

170

## Planned comparisons of means (ANOVA)

### comparison between means planned during study design, before data is obtained; for comparing ONE group w/ control (only 2 means); not common

171

## Unplanned comparisons of means (ANOVA)

### comparisons to determine differences between all pairs of mean; more common; controls Type I error

172

## Planned comparison calculations (ANOVA)

###
like a 2-sample t-test

test statistic: t =(Ybar1 - Ybar2)/SE

SE= √ MSerror (1/n1 + 1/n2)

note that we use error mean square instead of pooled variance (as in a normal t-test)

df = N-k

t critical= t0.05(2), df

173

## Unplanned comparison of means (ANOVA)

### Tukey-Kramer

174

## why do you need to know what kind of data you have

### determines what kind of statistical test you an do

175

## left skew

###
mean < median

skew 'pulls' mean in direction of skew

176

## C.I. notation

### 95% CI: a < µ < b (units)

177

## accept null hypothesis

###
NEVER!!!

only REJECT or FAIL TO REJECT

178

## why do we choose alpha = 0.05

### it balances TIE and TIIE which are actually conceptual, since we don't know if Ho is actually true or not

179

## standard error or estimate

### standard deviation of its sampling distribution; measures precision of the estimate

180

## SD vs. SE

###
SD- SPREAD of a distribution, deviation from mean

SE- PRECISION of an estimate; SD of sampling distribution

181

## test statistics

### used to evaluate whether the data is reasonably expected under the Ho

182

## P-value

### probability of getting the data, or something more unusual, given Ho is true

183

## reject Ho if

###
p-value ≤ alpha

less than OR equal to

0.049, 0.05

184

## Steps in hypothesis testing

###
1. State Ho and Ha

2. Calculate test statistic

3. Determine critical value or P-value

4. Compare test statistic to critical value

5. Evaluate Ho using sig. level (and interpret)

185

## Type I error

### Reject Ho, given Ho true

186

## Type II error

### Do not reject Ho, given Ho is false

187

## If we reduce alpha

### P[Type I] decreases, P[Type II] increases

188

## Experimental design steps

###
1.Develop clear statement of research question

2.List possible outcomes

3.Develop experimental plan

4.Check for design problems

189

## How to minimize bias

### control group, randomization, blinding

190

## How to minimize sampling error

###
replication- lare n lowers noise

balance- lowers noise

blocking

191

## to avoid pseudoreplication

### check df- obviously if its huge something is wrong

192

## Tukey-Kramer

### for 3 means: three Y bars, three Ho's; Q distribution; 3 row table w/ group i, group y, difference in means, SE, test statistic, critical q, outcome (reject/do not)

193

## Q-distribution

### symmetrical, uses larger critical values to restrict Type I error; more difficult to reject null

194

## Tukey-Kramer test statistic

###
q = Y_i(bar) - Y_j(bar) / SE

SE = √ MSerror(1/n1 + 1/n2)

195

## Tukey-Kramer testing

###
test statistic, q-value

critical value, q_α,k,N-k

k = # groups

N = total # observations

196

## Tukey-Kramer assumptions

###
random samples

data normally distributed in each group

equal variances in all groups

197

## 2 Factor ANOVA

### 2 Factors = 3 Ho's: difference in 1 factor, difference in 2nd factor, difference in interaction

198

## If interaction is significant

### do not conclude that factor is not

199

## Interaction plots

###
y-axis: response variable

x-axis: one of 2 main factors

legend for: other of 2 main factors (different symbols or colors)

2 lines

200

## interpreting interaction plot, interaction

### lines parallel: no significance in interaction

201

## interpreting interaction plot, b (data not on x-axis)

### take average along each line and compare the 2 on the y-axis, if they are not close then they are significant

202

## interpreting interaction plot, a (data on x-axis)

### x-axis: take average between the 2 dots (for each level of a), compare on y-axis, if they are not close they are significant

203

## control groups in an observational/experimental study will

###
reduce bias

will not affect sampling error

204

## correlation ≠

### causation

205

## correlation

###
"r"- comparing 2 numerical variables, [-1,1], no units, always linear

quantify strength and direction of LINEAR relationship (+/-)

206

## how to calculate correlation

###
r = signal/noise

signal= deviation in x and y together for every point (multiply each deviation before summing)

207

## correlation Ho

### no correlation between interbreeding and number of pup surviving their first winter (ρ = 0)

208

## determining correlation

###
test statistic: r/SE_r

SE_r = √ (1-r^2) / (n-2)

df = n-2

critical: tα,(2),df

compare statistic w/ critical

209

## df

###
n - number of parameters you estimate

correlation- you estimate 2

mann whitney- 0 parameters

210

## stating correlation results

### be careful not to interpret-- no causation!

211

## understanding r

### easy to understand because of lack of units, however, can trick you into thinking comparable across studies- across studies need to limit ranges

212

## Attenuation bias

### if x or y are measured with error, r will be lower; with increasing error, r is underestimated; avoided by taking means of subsamples

213

## correlation and significance

###
statistically sig. relationships can be weak, moderate, strong

sig.– probability, if Ho is true

correlation– direction, strength of linear relationship

214

## weak, moderate, strong correlation

###
r = ±0.2 – weak

r = ±0.5 – moderate

r = ±0.8 – strong

215

## correlation assumptions

###
bivariate normality- x and y are normal

relationship is linear

216

## dealing with assumption violations (correlation)

###
histograms

transformations in one or both variables

remove outlier

217

## outlier removal

###
–need justification (i.e. data error)

–carefully consider if variation is natural

–conduct analyses w/ and w/o outlier to assess effect of removal

218

## natural variation, outliers

### is your n big enough to detect if that is natural variation in the data

219

## if outlier removal has no effect

### may as well leave it in!

220

## non-parametric Correlation

### Spearman's rank correlation; strength and direction of linear association btw ranks of 2 variables; useful for outlier data

221

## Spearman's rank correlation assumptions

###
random sampling

linear relationship between ranks

222

## Spearman's rank correlation

###
r_s: same structure as Pearson's correlation but based on ranks

r_s = [Σ(Ri-Rbar)(Si-Sbar)] / [ Σ(Ri-Rbar)^2Σ(Si-Sbar)^2 ]

223

## conducting Spearmans

### rank x and y values separately; each data point will have 2 ranks; sum ranks for each variable; n = # data pts.; divide each rank sum by n to get Rbar and Sbar; calculate r_s (statistic); calculate critical r_s(0.05,df)

224

## if 2 points have same rank (Spearman)

### average of that rank and skip rank before/after; w/o any ties, the 2 values on the bottom of r_s equation will be the same

225

## Spearman hypothesis

### ρ_s = 0, correlation = 0

226

## Spearman df

### df = n because no estimations are being made in ranking

227

## linear regression

###
–relationship between x and y described by a line

–line can predict y from

–line indicates rate of change of y with x

Y = a + bX

228

## correlation vs. regression

###
regression assumes x,y relationship can be described by a line that predicts y from x

corr.- is there a relationship

reg.- can we predict y from x

229

## perfect correlation

### r = 1, all points are exactly on the line– regression line fitted to that 'line' could be the exact same line for a non-perfect correlation

230

## rounding mean results

### DO NOT; 4.5 puppies is a valid answer

231

## best line of fit

### minimizes SS = least squares regression; smaller sum of square deviations

232

## used for evaluating fit of the line to the data

### residuals

233

## residuals

### difference between actual Y value and predicted values for Y (the line); measure scatter above/below the line

234

## calculating linear regression

### calculate slope using b = formula; find a– a = Ybar - bXbar; plug in to Ybar = a + bXbar; rewrite as Y = a + bX; rewrite using words

235

## Yhat

### predicted value- if you are trying to predict a y value after equation has been solved

236

## why do we solve linear regression with Xbar, Ybar

### line of fit always goes through Xbar, Ybar

237

## how good is line of fit

###
MSresiduals = Σ(Yi - Yhat)^2 / n-2

which is SSresidual / n-2

quantifies fit of line- smaller is better

238

## Prediction confidence, linear regression

###
precision of predicted mean Y for a given X

precision of predicted single Y for a given X

239

## Precision of predicted mean Y for a given X, linear regression

### narrowest near mean of X, and flare outward from there; confidence band– most confident in prediction about the mean

240

## precision of predicted single Y for a given X, linear regression

### much wider because predicting a single Y from X is more uncertain than predicting the mean Y for that X

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

271

## design feature for investigating multiple variables and their interaction

### factorial design

272

## experiments with blocking

###
account for extraneous variation by putting experimental units into blocks that share common features

ex. instead of comparing randomly dispersed diversity, look at response variable within a block

273

## GLM, blocking

###
Ho: mean prey diversity is same in every fish abundance treatment

Ho: Diversity = grand mean + block

Ha: mean prey diversity is not the same in every fish abundance treatment

Ha: diversity = grand mean + block + fish abundance

274

## ANOVA, GLM, blocking

###
source of var.: block, abundance, residual, total

SS, df, MS, F, P

275

## Blocking Ho

###
Ho: mean prey diversity is the same in each block

Ha: mean prey diversity is not the same in each block

Block R^2 = SSblock / SStotal

Abundance + block R^2 =

SSabun. + SSblock / SStotal

276

## block as a variable

### block is an explanatory variable even if we are not inherently interested in its effect b/c it contributes to variation

277

## covariates

### reduce confounding variables, reduce bias

278

## ANCOVA, GLM

### Response = constant + explanatory + covariate

279

## ANOCVA hypotheses

###
Ho:No interaction between caste and body mass

Response = constant + exp. + covariate

Ha: Interaction between caste and body mass

Response = cons. + exp + cov. + explanatory*covariate

280

## ANCOVA hypotheses graphs

###
Ho: parallel

Ha: not parallel

affect is measured as the vertical difference between the two lines

281

## Testing ANCOVA

###
are the slopes equal

if not significant, drop interaction term and run model again

282

## df of interaction =

### df_covariant * df_explanatory

283

## Factorial design

###
multiple explanatory variables

fully factorial- every level of every variable and interaction is studied

284

## Factorial GLM statements

###
Ha: algal cover = grand mean + herbivory + height + herbivory*height

Ho: a.c. = G.M. + Herb. + Height

285

## GLM null hypotheses

###
do not include interaction statements

always one term different from alternative

286

## GLM degrees of freedom

###
explanatory:

df = levels of treatment - 1

interaction:

df = df_exp.1 * df_exp.2

df always total to grand n - 1

287

## Factorial GLM hypotheses graphs

###
Ho: no interaction = parallel lines

Ha: interaction = non parallel, maybe crossing lines

288

## Probability of independent events

###
P[X] = P[A]*P[B]*P[C]*....

if multiple ways to arrive at P[X] then add them up, or use Binomial (if conditions met)

289

## Binomial distribution

### probability distribution for # of successes in a fixed n of independent trials

290

## Binomial conditions

###
independent

probability of success is same for each trial

2 possible outcomes- success/failure

291

## proportion equations

###
p^ = X/n

SE_p^ = √ [p^ (1-p^)] / [n–1]

292

## Binomial test, testing proportions

###
whether relative frequency of successes in a population matches null expectation

Ho: p = p_o

293

## law of large numbers

### higher n = better estimate of p (or any estimate for that matter), lower SE

294

## binomial testing proportions calculations

###
test statistic = observed number of successes

null expectation = null 'p' * number of 'trials' (weighted by trials)

295

## steps in finding binomial p-value

### use null 'p' in binomial to calculate observed successes + anything more extreme; multiply by 2 (2 sided test)- this is the p-value; not comparing to critical value; compare to alpha

296

## binomial, p < 0.001

### reject Ho, p^ is significantly different than Ho: p = under a proportional model

297

## 95% CI for a population parameter

###
p' = ( X + 2 ) / ( n + 4 )

p' ± Z √ [p' (1–p')] / [n+4]

Z = 1.96 for 95% CI

298

## >2 possible categories

###
X^2 goodness-of-fit test

compare frequency data w/ >2 possible outcomes to frequencies expected from probability model in Ho

299

## Bar graphs

###
categorical data

space between bars

300

## X^2 example (days)

###
Ho: # of births is the same on each day

births on Monday is proportional to # of Mondays in the year

301

## X^2

### test statistic measures discrepancy btw observed (data) and expected (Ho) frequencies

302

## X^2 calculations

###
find E for each group, then X^2 for each group, sum X^2 = test statistic, compare to critical value

E = n*p

X^2 = Σ (O – E)^2 / E

df = # categories – 1

critical X^2_α,df

303

## Sampling distribution for Ho, binomial

###
Histogram- sampling distribution for all possible values for X^2

black line- theoretical X^2 probability distribution

304

## higher X^2 values

### observed farther from expected

305

## X^2, why -1 in df

### using n to calculate expected value- restricts data

306

## X^2 reject Ho

### data do not fit a proportional model, births are not equally distributed through the week

307

## X^2 goodness-of-fit assumptions

###
random sample

no category has expected frequency > 1

no more than 20% of the categories have expected frequencies < 5

308

## Poisson distribution

### describes probability of success in a block of time or space, when successes happen independently and with equal probability

309

## distribution of points in space

###
clumped

random

dispersed

310

## Poisson, P[X successes] =

###
E = e^-µ . µ^x / X!

µ = mean # of independent successes

311

## Poisson hypotheses

###
Ho: number of extinctions per time interval has a Poisson distribution

Ha: number of extinctions do not follow a Poisson distribution

312

## calculate a mean from a frequency table

### µ = (n1*f1)+(n2*f2)+(n3*f3)+.... / n

313

## hypothesis testing, poisson

###
calculate probability of success (expected value) for each level; calculate X^2 for each level, sum them; compare to critical value

df = # categories - 1

314

## determining if data are clumped or dispersed

###
s^2 =

[ Σ (Xi - µ)^2 * (obs. frequency)] / (n–1)

clumped: s^2 > µ

dispersed: s^2 < µ

315

## X^2 used for

###
proportional

binomial

poisson

316

## rejecting Ho, binomial

### probability of success is not same in all trials or trials are not independent

317

## rejecting Ho, poisson

### successes are not independent, probability of success is not constant over time or space

318

## contingency analysis

###
whether one variable depends on the other (is contingent on)

in a contingency table

explanatory variable in columns

response variable in row

each subject appears in table once

319

## contingency Ho

### no relationship between variables, variables independent

320

## associating categorical variables

###
test for association between ≥2 categorical variables

are categorical variables independent

odds ratio

X^2 contingency test

321

## odds ratio

###
to measure magnitude of association between 2 variables when each has only 2 categories

odds: O^ = p^ / 1–p^

odds ratio: OR = O1^ / O2^

322

## X^2 contingency test

### to test whether the 2 variables are independent; to test association between 2 categorical variables; need expected frequencies for each cell under Ho

323

## OR =

###
OR=1 : odds same for both groups

OR>1 : odds higher in 1st group- associated with increased risk

324

## expected frequencies, X^2 contingency

###
P[A ∩ B] =

(row total / grand total)(column total / grand total)

E = P[A ∩ B] * grand total

325

## calculating X^2 contingency

###
X^2 = Σ (O–E)^2 / E = test stat

df = (#rows–1)(#columns–1)

compare to critical value

326

## rejecting Ho, contingency

### Reject Ho that A and B are independent; P[A] is contingent upon B

327

## X^2 contingency test assumptions

###
random sample

no cells can have expected frequency <5

328

## if X^2 contingency test assumptions not met

### ≥2 rows/columns can be combined for larger expected frequencies

329

## to test independence of 2 categorical variables when expected frequencies are low

### Fisher's exact test

330

## Fisher's exact test

### gives exact p-value for a test of association in a 2x2 table

331

## Fisher's exact test assumptions

### random samples

332

## Fisher's Ho

### state of A and B are independent

333

## conduct Fisher's

###
–list all possible 2x2 tables w/ results as or more extreme than observed table

–p-value is sum of the Pr of all extreme tables under Ho of independence

–assess null

334

## Computer-Intensive methods

###
cheap speed

hypothesis testing- simulation, permutation (randomization)

standard errors, CI- bootstrapping

335

## hypothesis testing, simulation

###
–simulates sampling process many times- generate null distribution from simulated data

–creates a 'population' w/ parameter values specified by Ho

–used commonly when null distr. unknown

336

## simulation to generate null distribution

###
1.create and sample imaginary population w/ parameter values as specified by Ho

2.calculate test statistic on simulated sample

3.repeat 1&2 large number of times

4.gather all simulated test statistic values to form null distr.

5.compare test statistic from data to null distr. to approx. p-value and assess Ho

337

## generated null distribution

###
P-value ~ fraction of simulated X^2 values ≥ observed X^2

none ≥ observed, P < 0.0001

338

## Permutation tests (Randomization test)

### test hypotheses of association between 2 variables; randomization done w/o replacement; needs 'parameter' for association btw 2 variables

339

## Permutation test used when

### assumption of other methods are not met or null distribution is unknown

340

## Permutation steps

###
1.Create permuted data set w/ response variable randomly shuffled w/o replacement

2.calculate measure of association for permuted sample

3. repeat 1&2 large number of times

4. Gather all permuted values of test statistic to form null distribution

5. Determine approximate P-value and assess Ho

341

## Bootstrapping

###
calculate SE or CI for parameter estimate

useful if no formula or if distribution unknown

randomly 'resamples' from the data with replacement to estimate SE or CI

ex. median

342

## bootstrapping steps

###
1.random sample w/ replacement- 1st bootstrap sample

2.calculate estimate using bootstrap sample

3.repeat many times

4.calculate bootstrap SE

*only sampling from original sample values

343

## simulation

### mimics repeated sampling under Ho

344

## permutation

### randomly reassigns observed values for one of two variables

345

## bootstrapping

### used to calculate SE by resampling from the data set

346

## Jack-knifing

### leave-one-out method for calculating SE

347

## Jack-knifing

###
gives same result every time (unlike boot strapping)

calculates mean from n-1, then n-2, then n-3

348

## statistical significance

### observed difference (effect) are not likely due to random chance

349

## practical significance

### is the difference (effect) large enough to be important or of value in a practical sense

350

## effect size

###
ES– degree or strength of effect

ex. magnitude of relationship btw 2 variables

3 ways to quantify

351

## 3 ways to quantify ES

###
standardized mean difference

correlation

odds-ratio

352

## standardized ean difference

### Cohen's d

353

## can find statistical significance

### with a large n, which may not be large effect size, and may not be significant at lower n

354

## Quantifying ES

###
2% difference btw population and sample means

difficult to interpret mean differences w/o accounting for variance (s^2)

Cohen standardized ES w/ variance

355

## Cohen's d

###
simplest measure of ES

difference btw means / Sp

standardizes, puts all results on same scale (makes meta-analysis possible)

356

## Meta-analysis

###
analysis of analysis

synthesis of multiple studies on a topic that gives an overall conclusion; increases sig. of individual studies (larger n)

black line = 1-1 line - no difference, no more, no less

357

## steps in meta-anlysis

### define question to create one large study- general or specific; review literature to collect all studies- exhaustively; compute effect sizes and mean ES across al studies; look for effects of study quality

358

## literature search

### beware of 'garbage in, garbage out', publication bias, file-drawer problem

359

## publication bias

### bias- studies that weren't published- lower n, insignificant, low effect

360

## garbage in, garbage out

### justify why studies are not included, what is considered poor science?

361

## file-drawer problem

### studies that are not published- grad thesis, government research

362

## look for effects of study quality, Meta-analysis

###
do differences in n or methodology matter

-correlation btw n and ES?

-difference in observ. and exp. studies?

-base meta-analysis on higher quality studies

363

## pros of Meta-analysis

###
tells overall strength & variability of effect

can increase statistical power, reduce Type II error

can reveal publication bias

can reveal associations btw study type and study outcome

364

## cons/challenges of meta-analysis

###
assumes studies are directly comparable and unbiased samples

limited to accessible studies including necessary summary data

may have higher Type I error if publication bias is present

365

## what do we get out of the statistical process

###
a probability statement

this process is called Frequentist statistics, most commonly used

366

## What does frequentist statistics do

###
-answer probability statements if/given the null is true

-infer properties of a population using samples

-doesn't tell if null is true, not proof of anything

-useful, but must understand so not overinterpreted

367

## frequentists statistics developed

### Cohen, 1994; Null Hypothesis Sifnificance Testing

368

## why use frequentist statistics

###
appears to be objective and exact

readily available and easily used

everyone else uses it

scientists are taught to use it

supervisors & journals require it

369

## limits of frequentist statistics

###
–provides binary info only: significant or not

–does not provide means for assessing relative strength of support for alternate hypotheses

–failing to reject Ho does not mean Ho is true

–does not answer real question

370

## does not provide means for assessing relative strength of support for alternate hypotheses

### ex. conclude the slope of the line is not 0, how strong is the evidence that the slope is 0.4 vs 0.5

371

## real question

###
whether scientific hypothesis is true or false

-treatment has an effect (however small)

-if so, then Ho of no effect is false, but we are unable to show that Ho is false (or true)

-we can only show the probability of getting the data, if Ho is true

372

## question we CAN answer

### about the data, not the hypothesis- given the data, how likely is Ho to be true

373

## more limitations for frequentist stats

###
whether a result is significant depends on n, ES, alpha

significant does not always mean important

374

## larger n, ES, alpha

### increase likelihood of rejecting Ho- getting significant result

375

## significant does not necessarily mean important

### effects can be tiny and still statistically significant

376

## focus on p-values and Ho rejection

### distracts from the real goal- deciding whether data support scientific hypotheses and are practically/biologically important

377

## mostly we should be interested in

### size/strength/direction of an effect

378

## Bayesian statistics

### incorporate beliefs or knowledge of parameter values into analyses to contain population estimate

379