Topic 1 -8 Glossary symbols and Greek Alphabet for Moodle PDF

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UNSW PSYC3001 Glossary of notation, symbols and formulae – Dr Melanie Gleitzman

PSYC3001 Glossary – notation, symbols and formulae Below is a list of symbols (Roman symbols, mathematical symbols and Greek symbols) and formulae used in the course and their associated meaning. The Glossary is organised by Topic and lists symbols and formulae in the order they appear in the Lecture Slides. A table of symbols of the Greek Alphabet is given at the end. Symbol, notation or expression Topic 1 J

In words

Meaning or Formula

J

Number of groups (levels) of a single factor between-subjects design

N

N (upper case)

Number of participants in experiment

n

n (lower case)

Number of participants (observations) per group

nj

nj

Number of participants in group j

Yij

Y i, j

Score on dependent variable for ith participant in jth group, where i = 1, 2, ..., n and j = 1,2.

j

mu j

Population mean for jth level of factor

TE

treatment effect

Difference between population means, 1 – 2 sample mean for jth group

Mj

Mj

M1 – M2

Mean 1 minus Mean 2 point estimate of TE



sigma (uppercase)

Summation sign (mathematical operator)

SSj

Sum of squares for group j

Within group sums of squares,

SS

pooled SS

eg SS = SS1 + SS2

df

degrees of freedom

n1 + n2 – 2. When n1 = n2, df = 2n-2 = N - 2



sigma (lowercase)

population standard deviation

SE or ˆ M 1 M 2

estimated Standard error

ˆM1M 2 

eg SS1 = (Yi1 – M1)2

SS1  SS2  1 1     df  n1 n2 

when n1 = n2, ˆM1 M 2 

ˆ 2

estimate of population variance

ˆ 2 

SS1  SS2  2  df  n 

SS1  SS2 ie pooled SS divided by df df 1

UNSW PSYC3001 Glossary of notation, symbols and formulae – Dr Melanie Gleitzman

Symbol, notation or expression NHST

In words

Meaning or Formula

NHST

Null hypothesis significance test

H0

null hypothesis

eg H0: 1 - 2 = 0

H1

alternative hypothesis

eg H1: 1 - 2  0 (two-tailed)

t

t statistics

eg t 

/t/

absolute value of t

tc

t critical

M1  M 2 SE

critical value of t statistic, as a function of  and df eg tc = t/2, df



alpha

CI

confidence interval

100(1-)%

level of confidence

nominal Type I error rate (tests) or nominal non-coverage error rate (CIs) eg  = .05, 100(1-.05)% = (100 x .95)% = 95%



‘is an element of’

implies parameter is contained within CI limits

CI limits

confidence interval limits

eg 1   2  M1  M2    t /2 ;N  2 ˆM1 M2 

ll

lower limit of CI

in general, parameter  (ll, ul)

ul

upper limit of CI

Topic 2 PCER

per-comparison error rate

Per comparison type I error rate

FWER

familywise error rate

familywise type I error rate

Mmax – Mmin

M max minus M min

maximal comparison

Topic 3 M max  M min

range statistic

2 n

SSE

Sums of Squares Error

Within group sums of squares pooled across J groups, eg SS1 + SS2 + SS3 + ... + SSJ



‘nu’, pooled withingroups df

J(n – 1) = N – J

2

UNSW PSYC3001 Glossary of notation, symbols and formulae – Dr Melanie Gleitzman

Symbol, notation or expression MSE

In words

Meaning or Formula

Mean Square Error

pooled SS/pooled df = SSE/ = SSE / J(n-1)

q

q statistic

M j M j

max

MSE/n q*

q asterisk (or q star) statistic

M j M j ˆM

max



M j  M j

j  M j

MSE 

max

2 n

q ; J , 

critical value of q*

HSD

Honestly Significant Difference

( q*.05;4,80  SE), where SE  MSE 

CC

Critical constant

critical value for calculating CI limits

Tukey CC

q; J , 

Tukey CI limits



j

  j' 



M

j

2 n

 M j'   (q *.05; J,   MSE 

2 n

Topic 4 

mu

grand mean

j

alpha j

effect parameter for jth level of Factor A, j = j - 

ij

epsilon i j

error component for ith participant in jth group, ij = Yij - j

H0

homogeneity hypothesis

H0: 1 = 2 = ... = J = 0 or H0: 1 = 2 = ... = J

SSB

sums of squares between groups

SSB  n (M j  M )2

sums of squares error or sums of squares within groups

SSE  (Yij  M j )2

1

nu 1

df between groups, 1 = J - 1

2

nu 2, df error

df within groups (df error) 2 = J(n – 1) = N J

MSE or MSW

mean square error or mean square within

MSE 

SSE or SSW

j

i

j

SSE 2

3

UNSW PSYC3001 Glossary of notation, symbols and formulae – Dr Melanie Gleitzman

Symbol, notation or expression MSB

In words

Meaning or Formula

mean square between

MSB 

2

population error variance

ANOVA F

Analysis of variance F statistic

F

Fc

F critical

Fc = F.05, 1, 2



delta, non-centrality parameter



SSB 1

MSB MSE

n 2j j

(a function of SS of effect 2 parameters)

Topic 5  or g

psi or psi subscript g

population contrast (definition and value) or gth contrast

 g  c gj  j  c g 1 1  c g 2 2 

 or

 g   cgj  j  c g 11  c g 2  2 



j

j

c’g

coefficient vector

cg  c g 1 c g 2

cgj

contrast coefficient

coefficient for gth contrast for jth group

ˆ g

psi hat, g

sample value of gth contrast

cg 3

ˆ g   cgj M j  cg 1 M1  cg 2 M2 

J

j

ˆ ˆ

g

standard error of contrast

SE  ˆ ˆ g  MSE 

H0

contrast null hypothesis

H 0 : g  0

SS(ˆ)

Sums of squares of a contrast

SS(  ˆ) 

 cj2 n

ˆ )2 n ( c 2j j

 SS( )

Fˆ 

F statistic for a contrast

F ˆ 

CI limits

CI limits of a contrast

1  ˆ1   t/2; ˆˆ 

ˆ MSE

2

4

UNSW PSYC3001 Glossary of notation, symbols and formulae – Dr Melanie Gleitzman

Symbol, notation or expression d

In words

Meaning or Formula

Cohen’s d (standardised effect size)

d 

cgj+

sum of positive coefficients (across j groups) for gth contrast.

Mean difference scaling, c1j+ = 1.

t ˆ

t statistic for a contrast

t ˆ 

1   2 

ˆ ˆ  ˆ ˆ SE

Topic 6 STP

simultaneous test procedure

SCI

simultaneous confidence interval

F ˆ critical or F ˆ c    

Contrast Fc

Critical value of contrast F statistic.

CC

Critical constant

CC  F(ˆ )c

1  F ;1 ,2

Scheffé contrast Fc SSc

Sums of squares critical

1 × F; 1, 2 × MSE

Scheffé SSc Scheffé CC

Critical value of contrast SS.

Scheffé critical constant

CC  1 F ; , 1 2

 g  ˆ g  ( 1 F ;1 , 2  ˆˆ )

Scheffé SCI limits

g

ˆ max 

psi max

Coefficients of ˆ max 

maximal contrast Mj - M

Topic 7 k

lower case k

number of planned contrasts

EFER

expected family error rate

expected number of Type I errors

FWER < EFER

Bonferroni inequality

Familywise error rate (a probability) will be smaller than the expected number of Type I errors 5

UNSW PSYC3001 Glossary of notation, symbols and formulae – Dr Melanie Gleitzman

Symbol, notation or expression Bonferroni contrast Fc

In words

Meaning or Formula

Bonferroni SSc

Bonferroni sums of squares critical

F

Bonferroni CC

Bonferroni critical constant

CC  F / k ;1 , 

F

 / k ;1, 2

/ k ;1, 2

 MSE

2

 g  ˆ g  ( F / k;1,2  ˆˆ )

Bonferroni SCI limits

g

Topic 8 MCP

multiple comparison procedure

LSD

least significant difference procedure

‘protected’ t-test method

SNK

Student-Newman Keuls procedure

to be avoided at all costs

‘protection’

belief that preceding MCP with overall test will counter inflation of Type I error rate.

incoherent analysis

following a non-significant overall test with significant follow-up tests (based on a more liberal statistical method).

Greek Alphabet Symbol (upper case)

Symbol (lower case)

Name





alpha





beta





gamma





delta





epsilon





zeta





eta





theta





iota





kappa 6

UNSW PSYC3001 Glossary of notation, symbols and formulae – Dr Melanie Gleitzman

Symbol (upper case)

Symbol (lower case)

Name





lambda





mu





nu





ksi





omicron





pi





rho





sigma





tau





upsilon





phi





chi





psi





omega

7...