STATISTIEK II
Inhoud
Hoofdstuk 0 (proglegomena)................................................................................................................................................................1
statistiek, psychometrie en methodologie.......................................................................................................................................1
variabelen..........................................................................................................................................................................................1
Meetniveau’s.....................................................................................................................................................................................1
absoluut.........................................................................................................................................................................................1
Ratio..............................................................................................................................................................................................1
interval..........................................................................................................................................................................................1
ordinaal.........................................................................................................................................................................................1
Nominaal.......................................................................................................................................................................................1
Meetniveaus algemeen.....................................................................................................................................................................1
Zinvolheid (van een bewering)..........................................................................................................................................................2
Hoofdstuk 1 Data manipulatie..............................................................................................................................................................2
Basis commando’s in R-studio..........................................................................................................................................................2
Hoofdstuk 2 Beschrijvende statistiek....................................................................................................................................................3
Ordeningstechnieken........................................................................................................................................................................3
Grafische voorstellingen...................................................................................................................................................................3
Reductietechnieken..........................................................................................................................................................................3
gemiddelde ()................................................................................................................................................................................4
mediaan (mdx)...............................................................................................................................................................................4
Modus (mo)...................................................................................................................................................................................4
variantie (......................................................................................................................................................................................4
STANDAARDDEVATIE (Sx)..............................................................................................................................................................4
VARIATIEBREEDTE.........................................................................................................................................................................4
Interkwartiele afstand (q).............................................................................................................................................................4
spreidingsmaat d...........................................................................................................................................................................5
Covariantie (covxy).........................................................................................................................................................................5
correlatiecoëfficiënt van pearson (rXY)..........................................................................................................................................5
correlatiecoëfficiënt van Kendall’s tau (T)....................................................................................................................................5
regressielijn...................................................................................................................................................................................5
Hoofdstuk 3 Kansrekenen.....................................................................................................................................................................6
Toevalsvariabelen..............................................................................................................................................................................6
Kansverdelingen................................................................................................................................................................................6
univariate kansverdeling discrete variabelen...............................................................................................................................6
univariate dichtheidsfunctie continue variabele..........................................................................................................................6
, Bivariate kansverdeling discrete variabelen.................................................................................................................................7
bivariate dichtheidsfunctie continue variabele............................................................................................................................7
Centrum- en spreidingsmaten discrete variabelen...........................................................................................................................7
verwachting (E(x)).........................................................................................................................................................................7
populatievariantie (V(X) of σ2).......................................................................................................................................................7
populatie Standaarddeviatie (σ)...................................................................................................................................................7
associatietechnieken voor lineaire verbanden tussen discrete variabelen......................................................................................9
covariantie (COV(X, Y))..................................................................................................................................................................9
correlatiecoëfficiënt van pearson (...............................................................................................................................................9
Nuttige stellingen..............................................................................................................................................................................9
Centrale limietstelling.......................................................................................................................................................................9
De normaliteitsassumptie.................................................................................................................................................................9
De binomiale verdeling (discrete variabelen)...................................................................................................................................9
de normale verdeling (continue variabelen)...................................................................................................................................11
student verdeling = t-verdeling (continue variabelen)...................................................................................................................11
F-verdeling (continue variabelen)...................................................................................................................................................13
steekproefverdelingen....................................................................................................................................................................13
Gemiddelde van X.......................................................................................................................................................................13
Variantie van X............................................................................................................................................................................13
De steekproefverdeling van............................................................................................................................................................13
Verwachting van..........................................................................................................................................................................13
Variantie van...............................................................................................................................................................................13
Hoofdstuk 4 Puntschatting..................................................................................................................................................................14
eigenschappen van een goede schatter..........................................................................................................................................14
Enkele schatters..............................................................................................................................................................................14
schatter van de verwachtiing van variabele x.............................................................................................................................14
schatter van de variantie van variabele x...................................................................................................................................14
schatter van covariantie van X en Y............................................................................................................................................14
schatter van correlatiecoëfficiënt van pearson van X en Y.........................................................................................................14
Hoofdstuk 5 intervalschatting – betrouwbaarheidsintervallen..........................................................................................................15
opstellen betrouwbaarheidsinterval...............................................................................................................................................15
andere betrouwbaarheidsintervallen.............................................................................................................................................15
Hoofdstuk 6 de statistische toetsen....................................................................................................................................................16
de toetsingsprocedure....................................................................................................................................................................16
1) bepaal de theoretische hypotheses........................................................................................................................................16
2) Vertaal de theoretische hypothese in een alternatieve hypothese ()....................................................................................16
3) bepaal de nulhypothese ()......................................................................................................................................................16
4) controleer of je alternatieve hypothese wel zin heeft om te onderzoeken...........................................................................16
, 5) berekenen de toetsingsgrootheid g........................................................................................................................................17
6) bereken de p-waarde (bij een eenzijdige toets).....................................................................................................................17
7) bereken de p-waarde (bij een tweezijdige toets)...................................................................................................................17
8) beslis welke hypothese je verwerpt.......................................................................................................................................17
Voorbeeld........................................................................................................................................................................................17
t-toets (1 verwachting)....................................................................................................................................................................17
welch t-toets (2 verwachtingen).....................................................................................................................................................19
exacte binomiale toets (proporties)...............................................................................................................................................19
de normale quantile-quantile plot..................................................................................................................................................20
De significantie................................................................................................................................................................................20
De fouten.........................................................................................................................................................................................20
Hoofdstuk 7 De power........................................................................................................................................................................21
Power berekenen bij t-toets (1 verwachting).................................................................................................................................21
Power berekenen bij welch t-toets (2 afhankelijke verwachtingen)..............................................................................................21
Power berekenen bij welch t-toets (2 onafhankelijke verwachtingen)..........................................................................................21
power berekenen bij exacte binomiale toets (proporties).............................................................................................................22
algemeen.........................................................................................................................................................................................22
Hoofdstuk 8 enkelvoudige lineaire regressie......................................................................................................................................23
regressielijn berekenen...................................................................................................................................................................23
het enkelvoudig lineair model........................................................................................................................................................23
voorwaardelijke verwachting enkelvoudig lineair model...........................................................................................................24
voorwaardelijke variantie enkelvoudig lineair model................................................................................................................24
Verband correlatiecoëfficiënt en bij enkelvoudig lineair model.................................................................................................24
gauss-markov assumpties...............................................................................................................................................................24
de validiteit van de Gauss-Markov assumpties (adhv visuele analyse)..........................................................................................25
Gaus-markov assumptie 1...........................................................................................................................................................25
gaus-markov assumptie 2...........................................................................................................................................................25
puntschatting..................................................................................................................................................................................25
puntschatting van predictie........................................................................................................................................................25
puntschatting van predictie........................................................................................................................................................25
algemeen bij puntschatting predicties en...................................................................................................................................26
schatter........................................................................................................................................................................................26
intervalschatting..............................................................................................................................................................................26
tweezijdig betrouwbaarheidsinterval voor met betrouwbaarheid 1 –......................................................................................26
tweezijdig betrouwbaarheidsinterval voor met betrouwbaarheid 1 –......................................................................................27
.....................................................................................................................................................................................................27
algemene info toetsing...................................................................................................................................................................28
toetsen lineair model via de t-verdeling.........................................................................................................................................28
, toetsen lineair model via de F-verdeling........................................................................................................................................28
ssmod + SSres = SStot.............................................................................................................................................................................28
de determinatiecoefficient..............................................................................................................................................................30
de aangepaste determinatiecoëfficiënt......................................................................................................................................30
de R functie summary.....................................................................................................................................................................32
de power berekenen van de toets van...........................................................................................................................................32
relevante waarde bepalen..........................................................................................................................................................32
pwr.r.test.....................................................................................................................................................................................32
Hoofdstuk 9 meervoudige lineaire regressie......................................................................................................................................33
enkelvoudige vs meervoudige lineaire regressie............................................................................................................................33
visuele analyse................................................................................................................................................................................33
twee predictoren.........................................................................................................................................................................33
meer dan twee predictoren........................................................................................................................................................33
het meervoudig model....................................................................................................................................................................34
voorwaardelijke verwachting meervoudig lineair model...........................................................................................................34
voorwaardelijke variantie meervoudig lineair model.................................................................................................................34
Verband correlatiecoëfficiënt en bij meervoudig lineair model.................................................................................................34
puntschatting..................................................................................................................................................................................36
puntschatting van........................................................................................................................................................................36
puntschatting van........................................................................................................................................................................36
puntschatting van........................................................................................................................................................................36
collineariteit....................................................................................................................................................................................36
intervalschatting..............................................................................................................................................................................36
toetsing van meervoudige lineaire modellen.................................................................................................................................37
minstens 1 coëfficiënt van ≠ 0.....................................................................................................................................................37
Alle coëfficiënten van ≠ 0............................................................................................................................................................37
2 modellen vergelijken....................................................................................................................................................................38
selectie van een optimale subset van predictoren.........................................................................................................................38
Achterwaartse selectie/eliminatie..............................................................................................................................................39
voorwaartse selectie...................................................................................................................................................................41
stapsgewijze selectie...................................................................................................................................................................41
de determinatiecoëfficiënt R².........................................................................................................................................................42
de power van meervoudige lineaire regressie................................................................................................................................42
power van de f toets met p – k vrijheidsgraden.........................................................................................................................42
power van toets waarbij alle regressiecoëfficiënten = 0............................................................................................................42
power van toets waarbij regressiecoëfficiënt = nul....................................................................................................................43
validiteit van de gauss-markov assumpties....................................................................................................................................43
assumptie 1 gauss-markov..........................................................................................................................................................43
Inhoud
Hoofdstuk 0 (proglegomena)................................................................................................................................................................1
statistiek, psychometrie en methodologie.......................................................................................................................................1
variabelen..........................................................................................................................................................................................1
Meetniveau’s.....................................................................................................................................................................................1
absoluut.........................................................................................................................................................................................1
Ratio..............................................................................................................................................................................................1
interval..........................................................................................................................................................................................1
ordinaal.........................................................................................................................................................................................1
Nominaal.......................................................................................................................................................................................1
Meetniveaus algemeen.....................................................................................................................................................................1
Zinvolheid (van een bewering)..........................................................................................................................................................2
Hoofdstuk 1 Data manipulatie..............................................................................................................................................................2
Basis commando’s in R-studio..........................................................................................................................................................2
Hoofdstuk 2 Beschrijvende statistiek....................................................................................................................................................3
Ordeningstechnieken........................................................................................................................................................................3
Grafische voorstellingen...................................................................................................................................................................3
Reductietechnieken..........................................................................................................................................................................3
gemiddelde ()................................................................................................................................................................................4
mediaan (mdx)...............................................................................................................................................................................4
Modus (mo)...................................................................................................................................................................................4
variantie (......................................................................................................................................................................................4
STANDAARDDEVATIE (Sx)..............................................................................................................................................................4
VARIATIEBREEDTE.........................................................................................................................................................................4
Interkwartiele afstand (q).............................................................................................................................................................4
spreidingsmaat d...........................................................................................................................................................................5
Covariantie (covxy).........................................................................................................................................................................5
correlatiecoëfficiënt van pearson (rXY)..........................................................................................................................................5
correlatiecoëfficiënt van Kendall’s tau (T)....................................................................................................................................5
regressielijn...................................................................................................................................................................................5
Hoofdstuk 3 Kansrekenen.....................................................................................................................................................................6
Toevalsvariabelen..............................................................................................................................................................................6
Kansverdelingen................................................................................................................................................................................6
univariate kansverdeling discrete variabelen...............................................................................................................................6
univariate dichtheidsfunctie continue variabele..........................................................................................................................6
, Bivariate kansverdeling discrete variabelen.................................................................................................................................7
bivariate dichtheidsfunctie continue variabele............................................................................................................................7
Centrum- en spreidingsmaten discrete variabelen...........................................................................................................................7
verwachting (E(x)).........................................................................................................................................................................7
populatievariantie (V(X) of σ2).......................................................................................................................................................7
populatie Standaarddeviatie (σ)...................................................................................................................................................7
associatietechnieken voor lineaire verbanden tussen discrete variabelen......................................................................................9
covariantie (COV(X, Y))..................................................................................................................................................................9
correlatiecoëfficiënt van pearson (...............................................................................................................................................9
Nuttige stellingen..............................................................................................................................................................................9
Centrale limietstelling.......................................................................................................................................................................9
De normaliteitsassumptie.................................................................................................................................................................9
De binomiale verdeling (discrete variabelen)...................................................................................................................................9
de normale verdeling (continue variabelen)...................................................................................................................................11
student verdeling = t-verdeling (continue variabelen)...................................................................................................................11
F-verdeling (continue variabelen)...................................................................................................................................................13
steekproefverdelingen....................................................................................................................................................................13
Gemiddelde van X.......................................................................................................................................................................13
Variantie van X............................................................................................................................................................................13
De steekproefverdeling van............................................................................................................................................................13
Verwachting van..........................................................................................................................................................................13
Variantie van...............................................................................................................................................................................13
Hoofdstuk 4 Puntschatting..................................................................................................................................................................14
eigenschappen van een goede schatter..........................................................................................................................................14
Enkele schatters..............................................................................................................................................................................14
schatter van de verwachtiing van variabele x.............................................................................................................................14
schatter van de variantie van variabele x...................................................................................................................................14
schatter van covariantie van X en Y............................................................................................................................................14
schatter van correlatiecoëfficiënt van pearson van X en Y.........................................................................................................14
Hoofdstuk 5 intervalschatting – betrouwbaarheidsintervallen..........................................................................................................15
opstellen betrouwbaarheidsinterval...............................................................................................................................................15
andere betrouwbaarheidsintervallen.............................................................................................................................................15
Hoofdstuk 6 de statistische toetsen....................................................................................................................................................16
de toetsingsprocedure....................................................................................................................................................................16
1) bepaal de theoretische hypotheses........................................................................................................................................16
2) Vertaal de theoretische hypothese in een alternatieve hypothese ()....................................................................................16
3) bepaal de nulhypothese ()......................................................................................................................................................16
4) controleer of je alternatieve hypothese wel zin heeft om te onderzoeken...........................................................................16
, 5) berekenen de toetsingsgrootheid g........................................................................................................................................17
6) bereken de p-waarde (bij een eenzijdige toets).....................................................................................................................17
7) bereken de p-waarde (bij een tweezijdige toets)...................................................................................................................17
8) beslis welke hypothese je verwerpt.......................................................................................................................................17
Voorbeeld........................................................................................................................................................................................17
t-toets (1 verwachting)....................................................................................................................................................................17
welch t-toets (2 verwachtingen).....................................................................................................................................................19
exacte binomiale toets (proporties)...............................................................................................................................................19
de normale quantile-quantile plot..................................................................................................................................................20
De significantie................................................................................................................................................................................20
De fouten.........................................................................................................................................................................................20
Hoofdstuk 7 De power........................................................................................................................................................................21
Power berekenen bij t-toets (1 verwachting).................................................................................................................................21
Power berekenen bij welch t-toets (2 afhankelijke verwachtingen)..............................................................................................21
Power berekenen bij welch t-toets (2 onafhankelijke verwachtingen)..........................................................................................21
power berekenen bij exacte binomiale toets (proporties).............................................................................................................22
algemeen.........................................................................................................................................................................................22
Hoofdstuk 8 enkelvoudige lineaire regressie......................................................................................................................................23
regressielijn berekenen...................................................................................................................................................................23
het enkelvoudig lineair model........................................................................................................................................................23
voorwaardelijke verwachting enkelvoudig lineair model...........................................................................................................24
voorwaardelijke variantie enkelvoudig lineair model................................................................................................................24
Verband correlatiecoëfficiënt en bij enkelvoudig lineair model.................................................................................................24
gauss-markov assumpties...............................................................................................................................................................24
de validiteit van de Gauss-Markov assumpties (adhv visuele analyse)..........................................................................................25
Gaus-markov assumptie 1...........................................................................................................................................................25
gaus-markov assumptie 2...........................................................................................................................................................25
puntschatting..................................................................................................................................................................................25
puntschatting van predictie........................................................................................................................................................25
puntschatting van predictie........................................................................................................................................................25
algemeen bij puntschatting predicties en...................................................................................................................................26
schatter........................................................................................................................................................................................26
intervalschatting..............................................................................................................................................................................26
tweezijdig betrouwbaarheidsinterval voor met betrouwbaarheid 1 –......................................................................................26
tweezijdig betrouwbaarheidsinterval voor met betrouwbaarheid 1 –......................................................................................27
.....................................................................................................................................................................................................27
algemene info toetsing...................................................................................................................................................................28
toetsen lineair model via de t-verdeling.........................................................................................................................................28
, toetsen lineair model via de F-verdeling........................................................................................................................................28
ssmod + SSres = SStot.............................................................................................................................................................................28
de determinatiecoefficient..............................................................................................................................................................30
de aangepaste determinatiecoëfficiënt......................................................................................................................................30
de R functie summary.....................................................................................................................................................................32
de power berekenen van de toets van...........................................................................................................................................32
relevante waarde bepalen..........................................................................................................................................................32
pwr.r.test.....................................................................................................................................................................................32
Hoofdstuk 9 meervoudige lineaire regressie......................................................................................................................................33
enkelvoudige vs meervoudige lineaire regressie............................................................................................................................33
visuele analyse................................................................................................................................................................................33
twee predictoren.........................................................................................................................................................................33
meer dan twee predictoren........................................................................................................................................................33
het meervoudig model....................................................................................................................................................................34
voorwaardelijke verwachting meervoudig lineair model...........................................................................................................34
voorwaardelijke variantie meervoudig lineair model.................................................................................................................34
Verband correlatiecoëfficiënt en bij meervoudig lineair model.................................................................................................34
puntschatting..................................................................................................................................................................................36
puntschatting van........................................................................................................................................................................36
puntschatting van........................................................................................................................................................................36
puntschatting van........................................................................................................................................................................36
collineariteit....................................................................................................................................................................................36
intervalschatting..............................................................................................................................................................................36
toetsing van meervoudige lineaire modellen.................................................................................................................................37
minstens 1 coëfficiënt van ≠ 0.....................................................................................................................................................37
Alle coëfficiënten van ≠ 0............................................................................................................................................................37
2 modellen vergelijken....................................................................................................................................................................38
selectie van een optimale subset van predictoren.........................................................................................................................38
Achterwaartse selectie/eliminatie..............................................................................................................................................39
voorwaartse selectie...................................................................................................................................................................41
stapsgewijze selectie...................................................................................................................................................................41
de determinatiecoëfficiënt R².........................................................................................................................................................42
de power van meervoudige lineaire regressie................................................................................................................................42
power van de f toets met p – k vrijheidsgraden.........................................................................................................................42
power van toets waarbij alle regressiecoëfficiënten = 0............................................................................................................42
power van toets waarbij regressiecoëfficiënt = nul....................................................................................................................43
validiteit van de gauss-markov assumpties....................................................................................................................................43
assumptie 1 gauss-markov..........................................................................................................................................................43
