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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

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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

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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

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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

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distribution of points in space

clumped
random
dispersed

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Poisson, P[X successes] =

E = e^-µ . µ^x / X!
µ = mean # of independent successes

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Poisson hypotheses

Ho: number of extinctions per time interval has a Poisson distribution
Ha: number of extinctions do not follow a Poisson distribution

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calculate a mean from a frequency table

µ = (n1*f1)+(n2*f2)+(n3*f3)+.... / n

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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

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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^

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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

frequentists vs. bayesian example

100 coin flips all give 95 heads, what is the probability that the next flip will be a head?
freq.- 50%
bay.- 95%