Stats Review Data Aalysis as a Decisio Makig Process I Levels of Measuremet NOIR (See Whitley, 001, pp. 350-351, for details) Nomial Categories with Names; Yes vs. No (do t ask sometimes vs. ever), Sex, Religio, Relatioship Status, Political Affiliatio, Experimetal Group Membership (Cotrol Group, Maipulatio Group, Compariso Group). Numbers represet groups (1 Female, Male). Order is arbitrary. Ordial Nomial Categories with a Logical Order; Class Rak, Height from tallest to shortest, Resposes o a Numerical Ratig Scale (1 strogly agree, 7 Strogly disagree). Itervals betwee umbers is ot stadard from uit to uit. Iterval Numerical scales with logically ordered uits that are equidistat, but the zero is artificial. E.g., Temperature i Cetigrade ad Fahreheit (Zero does ot represet a absece of temperature). Time of day (o 0 o clock). Caledar Dates. Caledar Years. Ratio Numerical scales with logically ordered uits that are equidistat ad have a true Zero (the zero represets a lack of that which is beig measured). E.g., Elapsed Time, Temperature i Kelvi (0 degrees Kelvi -73.15 degrees celcius), Legth, Mass. Because it uses a true zero, umerical values ca be used to defie ratios: 5 iches is five times more legth tha 1 ich. 10 iches is twice as log as 5 iches. - Ratio Level Measures are Quite Rare i Psychology Cotiuous Vs. Discrete Variables Discrete Variables Mutually Exclusive/Exhaustive Numerical Categories that ca t be broke dow i to fier uits (e.g., if sex is represeted by 1:male ad :female, there is o 1.5). All Nomial ad Ordial Variables are Discrete. However, may Ordial variables will be treated as cotiuous (e.g., the Numerical ratig scales are ofte averaged to form a sigle score which is treated as cotiuous). Cotiuous Variables Numerical systems where there are a ifiite umber of possible poits betwee each uit. Also the measuremets ca be broke dow ito fier uits (e.g., elapsed time : Years, Moths, Days, Hours, Miutes, Secods, Millisecods, Naosecods, etc..). II. Choosig your Statistics Kowig which statistic to use to test the relatioship betwee each variable depeds o the type of data you have (ad sometimes the type of questio you wat to aswer) -Note: This is ofte discussed with respect to the issue of statistical validity ad there are differet camps regardig the Appropriateess of certai levels of measuremet for specific statistics (Michell, 1986). Also give specific coditios most parametric statistics ca had o-parametric data. A. Sigle Discrete Variable Goodess of Fit X Allows us to test whether the group frequecies differ from chace patters (base rate frequecies : the frequecy istaces aturally occur i the eviromet). (df k -1, where k umber of groups) χ ( O E ) i i E i where O i observed frequecy for each separate group E i expected group frequecies (based o chace)
Statistical Hypotheses: Ho : O f E f Ha : O f E f Example Research Questio: Does umber of people who say they like cheezy poofs (Yes 1) vs. those who do ot like cheezy poofs (No 0), differ sigificatly from the umber expected by chace aloe? B. Discrete X Discrete Pearso s X (AKA: Test of Idepedece) Allows us to test whether the cross tabulatio patter of two omial variables differs from the patters expected by chace. If oe variable is ordial the t or F are ormally used. (df (R-1)(C-1)) where R # of rows & C # of colums. χ ( O E ) ij ij E E ij ij RC i N j where i the differet groups for Variable 1 j the differet groups for Variable where Ri Row total of row i Cj Colum total of colum j Statistical Hypotheses: Ho : O f E f Ha : O f E f Example Research Questio: Does the umber of people who thik they are Eric Cartma (yes 1, o 0), relative to whether or ot they eat cheezy poofs, sigificatly differ from the frequecies that are expected by chace aloe. Note: a sigificat Pearso s chi square will ot tell you which cells are differet. The crosstabulatio matrix must be examied to determie this. Eye-ballig the stadardized, corrected residuals seems to be the most useful. Limitatios o X : 1) Resposes must be idepedet ad mutually exclusive ad exhaustive. Each case from the sample should fit ito oe ad oly oe cell of the cross tab matrix. ) Low expected Frequecies limit the validity of X. If df 1 (e.g., x matrix), the o expected frequecy ca be less tha 5. Also, If df, all expected frequecies should exceed. If df3 or greater, the all expected frequecies except oe should be 5 or greater ad the oe cell eeds to have a expected frequecy of 1 or greater. Phi Coefficiet (if X matrix) correlatio coefficiet that estimates the stregth of the relatioship betwee two dichotomous omial variables. Note: Phi ca ot estimate the directio (e.g., positive liear vs. egative liear) of the relatioship betwee omial variables because the umerical values are arbitrary (directio is meaigless). -This correlatio coefficiet ca be calculated exactly like Pearso s r (below) or ca be estimated usig the X statistic. Thus ay X ca be coverted to Phi or Phi ca be coverted to X. (sigificace should be determied usig X tables) φ χ N ad χ φ N
Statistical Hypotheses: Ho : Phi 0 Ha : Phi 0 Example Research Questio: What is the stregth of the relatioship betwee whether oe thiks they are Eric Cartma or ot (Yes 1, No 0) ad whether oe eats cheezy poofs (Yes 1, No 0). Cramer s Phi or V (If X3 matrix or larger) correlatio coefficiet that estimates the stregth of the relatioship betwee two discrete omial variables (correctig for the ifluece of the umber of groups). (Sigificace should be determied usig X tables). Note: Phi ca ot estimate the directio (positive vs. egative) of the relatioship betwee omial variables because the umerical values are arbitrary (directio is meaigless). φ cramer χ N( k 1) where k is the smaller of R or C (or whe r c, k r c) Statistical Hypotheses: Ho : Phi 0 Ha : Phi 0 Example Research Questio: What is the stregth of the relatioship betwee whether oe thiks they are Eric Cartma or ot (Yes 1, No 0) ad whether oe prefers cheezy poofs (1), HooHoo Dillies (), or coa-coa yum-yum s (3). - If both discrete variables are Ordial (Raked Data) Spearma s Rho r s : Correlatio coefficiet that estimates the stregth ad directio of the relatioship betwee two ordial variables. Calculated usig Pearso s r formula. However, the r table ca t be used to estimate sigificace. For N > 10 ad N < 8, covert to a Z score. If Z greater tha 1.96, the alpha <.05. If Z greater tha.85, the alpha <.01. For smaller ad larger N, respectively, special tables will eed to be obtaied. Z rs ( N + 5) 9N( N 1) Statistical Hypotheses: Ho : Rho 0 Ha : Rho 0 Example Research Questio: What is the stregth ad directio of the relatioship betwee Eric Cartma s rakigs of 0 participat s suitability as a mate (rage 1-0) ad participats rakigs with respect to how may Cheezy Poofs they eat per-week ( rage 1-0). Kedall s Tau J : This statistic provides a alterative to Rho that is easier to calculate sigificace for by had. Rather tha beig a variatio of Pearso s r, Tau is based i the umber of iversios betwee the raked items. For example, the followig data set cotais 4 iversios out of the 10 comparisos. Subject 1 3 4 5 6 7 8 9 10 Var 1 1 3 4 5 6 7 8 9 10 Var 1 5 3 4 6 7 9 8 10 Iversio Iversio Iversio Iversio
τ 1 ( I) N( N 1) where I umber of iversios Ofte, Tau is used as a measure of iter-rater agreemet (though Rho ca also be used for such purposes). Sigificace (regardless of the umber of subjects) is tested usig the same Z test as for Rho, oly substitute J for r s. Use the same Critical Values of Z: If Z greater tha 1.96, the alpha <.05. If Z greater tha.85, the alpha <.01. Statistical Hypotheses: Ho : Tau 0 Ha : Tau 0 Example Research Questio: What is the stregth ad directio of the relatioship betwee Eric Cartma s rakigs of 0 participat s suitability as a mate (rage 1-0) ad participats rakigs with respect to how may Cheezy Poofs they eat per-week ( rage 1-0). C. Discrete X Cotiuous If Discrete Variable is Dichotomous (oly levels) z-test compares a sigle sample mea to the populatio mea, whe the populatio stadard deviatio is kow Z X μ σ N where : & F populatio mea ad stadard deviatio, respectively Sigificace Use the Critical Values of Z: If Z greater tha 1.96, the alpha <.05. If Z greater tha.85, the alpha <.01. Statistical Hypotheses: Ho : Group 1 mea Populatio Mea Ha : Group 1 mea Populatio Mea Example Research Questio: Does the average umber of Cheezy Poofs eate per day by studets eroled i Graduate Research Statistics (rage 0-600) differ from the average umber of Cheezy Poofs eate per day i the geeral populatio of Graduate Studets. t-test Sigificace : If t obtaied exceeds the t critical (see ay t table for critical values) for a give df at the.05 alpha level, the the groups are sigificatly differet. Sigle Sample t-test compare a sample mea to a populatio mea whe the oly the sample stadard deviatio is kow. df -1 t X μ s Statistical Hypotheses: where s sample stadard deviatio Ho : Group 1 mea Populatio Mea Ha : Group 1 mea Populatio Mea Example Research Questio: Does the average umber of Cheezy Poofs eate per day by studets eroled i Graduate Research Statistics (rage 0-600) differ from the average umber of Cheezy Poofs eate per day i the Uited States.
Idepedet Sample t-test compare two the meas of two urelated groups. df - t X1 X s s 1 1 1 1 + + 1 1 ( 1) + ( 1) Statistical Hypotheses: Ho : 0 1 0 Ha : 0 1 0 Example Research Questio: Does a radomly assiged group exposed to 37 hours of South Park (group 1) rerus report sigificatly more positive or egative attitudes toward Research Methods (measured usig a 5 item questioaire employig a 7 poit ratig scale: item averages rage from 1-7) compared to a radomly assiged compariso coditio exposed to 37 hours of Sally Struthers Feed the Childre commercials (group ). Repeated Measure / Matched Sample t-test Repeated Measure: test the sigificace of the averaged differece i scores betwee time 1 ad time. Matched Sample: compare averaged differece i scores betwee group 1 ad group whe the subjects from each group have bee matched o some variable (e.g. age, itelligece, etc.). df -1 t dif X Σ ( X X ) t1 t Xt ( Σ ( Xt Xt )) 1 t1 1 Statistical Hypotheses: Ho : 0 t1 0 t Ha : 0 t1 0 t Example Research Questio: After beig exposed to 37 hours of South Park rerus (Time ), do participats report sigificatly more positive or egative attitudes toward Research Methods (measured usig a 5 item questioaire employig a 7 poit ratig scale: item averages rage from 1-7) compared to their pre-exposure scores (Time 1). Biserial vs. Poit Biserial (Artificial Dichotomies vs. Natural Dichotomies). Poit Biserial Correlatio Coefficiet r pb Same formula as Perso s r (see below), oly oe variable is a atural dichotomy ofte desigated by 0 & 1. This correlatio coefficiet will idicate the stregth of the relatioship betwee category membership ad the cotiuous score. Note that like Phi, the directio of the relatioship (positive vs. egative) is arbitrarily based o the umerical labels assiged to the groups. Examiatio of the meas is ecessary to determie the directio of the group differeces. df - Statistical Hypotheses: Ho : r pb 0 Ha : r pb 0 Example Research Questio: What is the stregth of the relatioship betwee beig radomly assiged to a group exposed to 37 hours of South Park rerus (group 1)
vs. beig radomly assiged to a compariso coditio exposed to 37 hours of Sally Struthers Feed the Childre commercials (group 0) ad self-report attitudes toward Research Methods (measured usig a 5 item questioaire employig a 7 poit ratig scale: item averages rage from 1-7). Biserial Correlatio Coefficiet r b Used whe a Dichotomy is developed from a cotiuous variable (e.g. mea split, or media split methods). Groups are ofte desigated by 0 & 1. r b is estimated from a ormal Pearso s r (see below). This statistic will tell you the stregth of the associatio betwee category membership (based o a artificial dichotomy) ad a cotiuous score. Agai, directio of the relatioship will be arbitrary depedig o the umerical category labels (however sice they are based o cotiuous scores the umerical label with greater value should be give to the upper ed of the cotiuum, makig iterpretatio easier) df - r b r % % X μ pearso belowcp abovecp cp σ where X cp the raw score cut poit (raw score used to split distributio ito groups) Statistical Hypotheses: Ho : r b 0 Ha : r b 0 Example Research Questio: What is the stregth ad directio of the relatioship betwee watchig more tha 10 hrs. per week of South Park (Group 1) vs. watchig 10 or fewer hours of South Park per week (group 0) ad self-report attitudes toward Research Methods (measured usig a 5 item questioaire employig a 7 poit ratig scale: item averages rage from 1-7), where 10 hrs. per week is the mea of selfreported South Park viewig habits. If Discrete Variable has more tha Levels (more tha groups). Oe Way ANOVA : Aova tests whether or more group meas are sigificatly differet. -Whe usig groups F t. -See ANOVA hadout for details o Oe Way Aova Statistical Hypotheses : Ho : Mea grp1 Mea grp Mea grp j (for j groups) Ha : At least oe group mea sigificatly differet from oe other group mea. Example Research Questio: Are there ay sigificat differece betwee three radomly assiged groups ( 1, exposed to 37 hours of South Park Episodes;, exposed to 37 hours of Sally Struthers Feed the Childre commercials; & 3, o TV cotrol coditio) with respect to their self-report attitudes toward Research Methods (measured usig a 5 item questioaire employig a 7 poit ratig scale: item averages rage from 1-7). If Discrete X Discrete X Cotiuous (Where both discrete variables are predictors) Two Way ANOVA : If we have idepedet variables (or 1 IV ad a Blockig Variable) the Two Way Aova (Or Factoral ANOVA) is called for. Agai, this will tell us if oe group mea (or matrix cell mea) is sigificatly differet from oe other group mea (or cell mea). Also, Factoral Aova ca hadle More tha IV s
(ad or Blockig Variables). (See Two Way Aova Hadout for Details). Moderatio - whe we fid a sigificat iteractio betwee two predictor variables we ca say that oe predictor moderates the relatioship betwee the other predictor ad the outcome (DV). Our decisio about which predictor is the IV ad which is the Moderatig Variable is based o our theoretical perspective. Statistical Hypotheses - The two way ANOVA actually tests several hypotheses at oce. 1) Mai Effects IV1 : Ho : Mea Group 1. Mea Group i. (for i groups) Ha : At least oe group mea sigificatly differet from oe other group mea. IV : Ho : Mea Group.1 Mea Group.j (for j groups) Ha : At least oe group mea sigificatly differet from oe other group mea. ) Iteractio effects (moderatio effects) IV1 : Ho : Mea grp11 Mea grp 1 Mea grp1 Mea grp ij (for i ad j groups) Ha : At least oe group mea sigificatly differet from oe other group mea. Example Research Questio: Do males (sex 0) have sigificatly more positive or egative attitudes toward Research Methods (measured usig a 5 item questioaire employig a 7 poit ratig scale: item averages rage from 1-7) compared to females (sex 1). Also, does a radomly assiged group exposed to 37 hours of South Park (group 1) rerus report sigificatly more positive or egative attitudes toward Research Methods (measured usig a 5 item questioaire employig a 7 poit ratig scale: item averages rage from 1-7) compared to a radomly assiged compariso coditio exposed to 37 hours of Sally Struthers Feed the Childre commercials (group ). Does participat sex (m 0, f 1) moderate the relatioship betwee south park exposure (IV: radom assigmet to coditio: 1) exposed to 37 hours of South Park Episodes; ) exposed to 37 hours of Sally Struthers Feed the Childre commercials.) ad attitude toward Research Methods (measured usig a 5 item questioaire employig a 7 poit ratig scale: item averages rage from 1-7). ANCOVA : Aalysis of Covariace : Sometimes we wat to remove the effects of a third variable (Covariate: Ca be cotiuous or discrete). We may wat to remove the effects of a uisace variable that is correlated with our mai variables of iterest or we may wat to test a mediatioal hypothesis (that the 3 rd variable explais the relatioship betwee the IV ad DV. that is, the 3 rd variable accouts for all the shared variace betwee the IV & DV). (see a good stats book for details) Statistical Hypotheses: Ho : Mea Group1 Mea Groupj (for j groups) Ha : At least oe group mea sigificatly differet from oe other group mea. Example Research Questio: Do males (Sex 0) have sigificatly more positive or egative attitudes toward Cheezy Poofs (measured usig a 6 item questioaire employig a 5 poit ratig scale: item averages rage from 1-5) compared to females (Sex 1) after the effects associated with IQ (rage 70-150) are removed. NOTE - Multiple Regressio : Aythig Aova ad Acova ca do, Multiple regressio ca do as well through the use of dummy codig, effects codig, ad cotrast codig.
r D. Cotiuous X Cotiuous 1. Pearso s r : Allows us to test the stregth of the associatio betwee two cotiuous variables. It represets a ratio of the Covariace (variace shared by two variables) ad the total variace (covariace + uique variace). df - ( ΣX)( ΣY) Σ XY ( Σ X ) ( Σ Y) Σ X Σ Y or r ( X X)( Y Y) ( X X) ( Y Y) Statistical Hypotheses: Ho : r 0 Ha : r 0 Example Research Questio: What is the stregth ad directio of the associatio betwee the umber of Viea Sausages that a perso eats i 4 hr period (measured i grams, rage 0-900) ad the amout of time they sped doig impersoatios of south park characters i that same 4 hr. period (measured i miutes, rage 0-1440). r : The Coefficiet of Determiatio : r represets a ratio of covariace to total variace, but if we wat to kow how much variace i the DV is explaied by variace i the IV, the we ca square r to fid out. NOTE : for all of the correlatio based statistics above (Rho, Tau, Biserial, Poit-Biserial) if the sample size exceeds 30 the the Pearso r is robust eough that it ca will provide a reasoable approximatio of ay of these statistics without sigificatly iflatig the Type I error rate. Sigificace of r : sigificace tables for r are available i most stats books. If r obtaied exceeds the critical value of r for a give df at the.05 alpha level, the there is a sigificat associatio betwee the variables of iterest. If r tables are ot available you ca use t tables, as r tables are based o t coversios. t 1 r. Partial Correlatio pr : Tests the associatio betwee two cotiuous variables with the effects of a third variable removed. We may wat to remove the effects of a uisace variable that is correlated with our mai variables of iterest or we may wat to test a mediatioal hypothesis (that the 3 rd variable explais the relatioship betwee the IV ad DV. that is, the 3 rd variable accouts for all the shared variace betwee the IV & DV). (see a good stats book for details) -Note: The covariate does ot ecessarily have to be cotiuous (especially if your N is larger tha 30). Statistical Hypotheses: Ho : pr 0 Ha : pr 0 Example Research Questio: What is the stregth ad directio of the associatio betwee the umber of Viea Sausages that a perso eats i 4 hr period (measured i grams, rage 0-900) ad the amout of time they sped doig impersoatios of south park characters i that same 4 hr. period (measured i miutes, rage 0-1440) whe the variace i attitudes toward Viea Sausages (measure usig 4 item
measure employig a 7 poit umerical ratig scale, item averages rag from 1 to 7) is removed. 3. Regressio: Whe you have a sigle variable, regressio is essetially the same as correlatio. Istead of r, you calculate for b (beta), where b reflects the slope of lie that best fits the data (least-squares regressio coefficiet). Stregth of associatio is represeted as the chage i Y (DV) that results from a 1 uit chage i X (IV). Equatio for a Straight Lie (Least-Squares Regressio) Y a + bx Where: Y Depedet Variable: see as a fuctio of (or predicted by) the idepedet variable (X); X Idepedet Variable: the dimesio or characteristic that is see as the determiat or cause of the depedet variable (Y); b Slope of the Lie: rise (or drop) divided by ru; a Y-itercept: where the value of X 0 ad the lie itercepts the Y axis. Formula for Calculatig a ad b b Σ XY Σ X ( ΣX)( ΣY) ( Σ X ) a y bx Oce a ad b are kow they ca be plugged ito the regressio lie formula (Y a + bx) ad the predicted value of Y ca be estimated for ay value of X. Example Research Questio- For a give slope (b) of.6300 ad a Y-itercept (a) of 3., the how much time would a perso be expected to sped impersoatig Southpark characters (measured i miutes, rage 0-1440) i 4 hour period after eatig 300 grams of Viea Sausages? E. Multiple Cotiuous Idepedet Variables ad Sigle Cotiuous Depedet Variable Multiple Regressio - Like Factorial Aova, Multiple Regressio ca deal with multiple Idepedet Variables. Also, as a form of regressio, multiple regressio ca be used for predictio (predictig Y based o the values of the IVs). R The Multiple Correlatio Coefficiet - idicates the total associatio betwee the Predictors (IVs) ad the Criterio (DV). R Squared Multiple Correlatio Coefficiet idicates the % of the variace i the DV that is accouted for by the IVs. F used to test the sigificace of R. b The multiple regressio coefficiet - idicates the icrease i Y resultig from a 1 uit icrease i Y, whe all other IVs are held at the costat of 0. That is, it tells us about the uique effect a give predictor has o the criterio $ Beta The stadardized multiple regressio coefficiet - same as b oly the uits are stadardized i Z-score uits.
t (b/stadard error for b) used to test the sigificace of the regressio coefficiet. - The research questios that you ca ask with Multiple regressio are quite flexible. - Sigle Step - idetifies the uique associatio of each predictor with a criterio - Hierarchical Regressio (Mulitple Steps) - idetifies the uique cotributio of a sigle variable or group of variables to the multiple correlatio coefficiet (R )) - Mediatio Aalyses - (Because) A third variable accouts for/explais the relatioship betwee X ad Y. Why is X related to Y, because of Z. -This is the ultimate goal of Sciece. - Moderatio Aalyses / Iteractio effects - (It Depeds) A third variable iflueces the stregth ad/or directio of the relatioship betwee a IV ad DV. What ifluece does X have o Y? It depeds o Z. F. Multiple Depedet Variables 1. Discrete IVs ad Multiple Cotiuous DV s MANOVA : Multivariate Aalysis of Variace - Whe you have multiple cotiuous outcome variables that reflect a related set of costructs ad you wat to test the associatio with 1 or more discrete IV s you ca use Multiple Aalysis of Variace. - Returs a sigle F that idicates whether the IV (or IVs) are sigificatly associated with the DVs as a group. - This is most useful for keepig the Type I error rate dow whe coductig multiple aalyses. - If it is sigificat the it is usually followed up with Uivariate tests assessig oe depedet variable at a time. - (see a good stats book for details) MANCOVA : Maova with a Covariate (a variable that is havig it s ifluece removed from the test). (see a good stats book for details). Oe or more Cotiuous Idepedet Variables ad Multiple Cotiuous Depedet Variables - Caoical Correlatio or Set Correlatio - Returs a sigle correlatio coefficiet that is the best fittig correlatio betwee set 1 (IVs) ad set (DVs) determied through multiple iteratios. - Path Aalysis / Structural Equatio Modelig - Theory/Model Testig Procedures - - Allows you to determie the degree to which causal relatioship predicted by theory fit with the data; Goodess of Fit, which is expressed as a chi-square ad other fit idicies. - Path Aalysis / Causal Modelig - oly cosiders Maifest Variables, which are directly measured variables.
- Structural Equatio Modelig / Latet Variable Models - iclude Latet Variables, which are ot measured directly. For example SES, is ot determied by a sigle idicator. It is a latet variable made up of maifest variables like icome, educatio, job prestige, ad obtaied wealth. Sample Structural Model G. Cotiuous Predictors ad Categorical Depedet Variables - Logistical Regressio - Allows you to ask a variety of research questios with miimal statistical assumptios (e.g., assumptios regardig ormal distributios). - The most importat of which would be Stregth of Associatio betwee predictors ad outcomes ad Predictio (predictig outcomes/group membership for future cases). - Logistic regressio ca hadle multiple predictors that either cotiuous or discrete ad combiatios of both.
G. Data Reductio - Reducig a larger umber of variables dow to more maageable set. - Selectig Items for scale developmet - Factor Aalysis / Priciple Compoet Aalysis - Exploratory - - Idetifies groups of variables that have bee respoded to i similar way whe o a priori groups have bee idetified. - Cofirmatory - - Similar to SEM ad Path Aalysis, idetifies the goodess of fit betwee priori groups of items/variables ad the data. - Cluster Aalysis - Like Factor aalysis but groups people istead of varaibles. I. There are others... May May Others... - Multi Level Modelig (for group level idepedet variables) - Hierarchical Liear Modelig (kid of a combiatio of cluster aalysis ad regressio)