Descriptive Methods 707.031: Evaluation Methodology Winter 2015/16 Eduardo Veas what we do with the data depends on the scales 2 Measurement Scales 3 The complexity of measurements • Nominal Crude • Ordinal • Interval • Ratio Sophisticated 4 Nominal data • arbitrarily assigning a code to a category or attribute: postal codes, job classifications, military ranks, gender • mathematical manipulations are meaningless • mutually exclusive categories • each category is a level 5• use: freq, counts, Ordinal data • ranking of an attribute • interval between points in scale not intrinsically equal • comparisons < or > are possible 6 Interval data • equal distances between adjacent values, but no absolute zero • temperature in C or F • mean can be computed • Likert scale data ? 7 Ratio • absolute zero • can be operated mathematically • time to complete, distance or velocity of cursor, • count, normalized count (count per something) 8 Frequencies Title Text 9 Frequency tables • tab.courses<- as.data.frame(freq(ordered(courses)), plot=FALSE) • CumFreq= cumsum(tab.courses[- dim(tab.courses)[1],]$Frequency) • tab.courses$CumFreq=c(CumFreq,NA) • tab.courses 10 Interpreting frequency tables Frequency Percent CumPercent CumFreq 1 2 20 20 2 2 3 30 50 5 3 4 40 90 9 4 1 10 100 10 Total 10 100 NA NA 11 Contingency Tables Right-handed Left-handed Total Males 43 9 52 Females 44 4 48 Totals 87 13 100 sd 12 Modelling 13 Statistical models • A model has to accurately represent the real world phenomenon. • A model can be used to predict things about the real world. • The degree to which a statistical model represents the data collected is called fit of the model 14 Frequency distributions • plot observations on the x-axis and a bar showing the count per observation • ideally observations fall symmetrically around the center • skew and kurtosis describe abnormalities in the distributions 15 Histogram / Frequency distributions 16 Center of a distribution • Mode: score that occurs most frequently in the dataset • it may take several values • it may change dramatically with a single added score • Median: is the middle score (after ranking all scores) • for even nr of scores, add centric values and divide by 2 • good for ordinal, interval and ratios • Mean: average score 17 • can be influenced by extreme scores Dispersion of a distribution • range: difference between lowest and highest score 252 - 22 = 232 121 - 22 = 99 • interquartile difference: mode + upper and lower quartiles 18 Fit of the mean • deviance: mean - x • sum of squared errors (SS) • variance = SS / N-1 • stddev = 19 sqrt(variance) Assumptions 20 Assumptions of parametric data • normally distributed: sample or error in the model • homogeneity of variance: • correlational: variance of one variable should be stable at all levels of the other variable • groups: each sample comes from a population with same variance • interval data: at least interval data • independence: the behaviour of one participant does not 21 influence that of another Distributions for DLF 1.2 0.75 0.6 0.9 0.50 0.4 0.6 Density Density Density 0.25 0.2 0.3 0.0 0.00 0.0 0 1 2 3 4 0 1 2 3 0 1 2 3 Hygiene score on day1 Hygiene score on day 2 Hygiene score on day 3 3 3 3 2 2 2 sample sample sample 1 1 1 0 22 0 0 -2 0 2 -3 -2 -1 0 1 2 3 -2 -1 0 1 2 theoretical theoretical theoretical Quantify normallity 23 Different groups 24 Exam histogram 0.025 0.020 0.015 density 0.010 0.005 0.000 25 50 75 100 25 exam Exam histogram 0.04 0.025 0.03 density 0.02 0.020 0.01 0.015 0.00 10 20 30 40 50 60 70 exam density 0.010 0.06 0.005 0.04 density 0.000 0.02 25 50 75 100 exam 0.00 26 60 70 80 90 100 exam Shapiro-Wilk test • # Shapiro-Wilk • shapiro.test(rexam$exam) • • #if we are comparing groups, what is important is the normallity within each group • by(rexam$exam, rexam$uni, shapiro.test) 27 Reporting Shapiro-Wilk • A Shapiro-Wilk test on the R exam, W=0.96, proved a significant deviation from normality (p<0.05). 28 Homogeneity of variance • Levene’s test: • leventTest(rexam$exam, rexam$uni, center=mean) • Reporting: for the percentage on the R exam, the variances were similar for KFU and TUG students, F(1,98)=2.09 29 Homogeneity of variance • Levene in large datasets may give sig for small variations • Double check Variance ratio (Hartley’s Fmax) 30 Correlations Title Text 31 Everything is hard to begin with, but the more you practise the easier it gets 32 Relationships • Everything is hard to begin with, but the more you practise the easier it gets • increase in practice, increase in skill • increase in practice, but skill remains unchanged • increase in practice, decrease in skill 33 Correlations • Bivariate: correlation between two variables • Partial: correlation between two variables while controlling the effect of one or more additional variables 34 Covariance • are changes in one variable met with similar changes in the other variable • cross product deviations= multiply deviations of the two variables • covariance= CPD / (N-1) 35 Covariance II • Positive: both variables vary in the same direction • Negative: variables vary in opposite directions • Covariance is scale dependent and cannot be generalized 36 Pearson correlation coefficient • cov/sxsy • Data must be at least interval • Value between -1 and 1 • 1 -> variables positively correlated • 0 -> no linear relationship • -1 -> variables negatively correlated 37 Dataset Exams and Anxiety • effects of exam stress and revision on exam performance • questionnaire to assess anxiety relating to exams (EAQ) 38 Enter data • examData<-read.delim("ExamAnxiety.dat", header=TRUE) • examData2<- examData[,c(“Exam”,"Anxiety","Revise")] • cor(examData2) 39 Pearson correlation • Exam Anxiety Revise • Exam 1.0000000 -0.4409934 0.3967207 • Anxiety -0.4409934 1.0000000 -0.7092493 • Revise 0.3967207 -0.7092493 1.0000000 40 Confidence values • rcorr(as.matrix(examData[,c(“Exam","Anxiety","R evise")])) • Exam Anxiety Revise • Exam 0 0 • Anxiety 0 0 • Revise 0 0 41 Reporting Pearson’s CC A Pearson correlation coefficient indicated a significant correlation between anxiety performance and time spent revising, r=-.44, p<0.01 42 Spearman’s correlation coefficient • non parametric test • first rank the data and then apply Pearson cc 43 Liar Dataset • contest for storytelling the biggest lie • 68 participants, ranking, and creativity questionnaire 44 Spearman test • liarData=read.delim("biggestLiar.dat", header=TRUE) • rcorr(as.matrix(liarData[,c(“Position","Creativity") ])) • Position Creativity • Position 1.00 -0.31 • Creativity -0.31 1.00 45 Reporting spearman A Spearman non-parametric correlation test indicated a significant correlation between creativity and ranking in the world’s biggest liar contest, r=-.37, p<0.001 46 Kendall’s tau non-parametric • used for small datasets • cor.test(liarData$Position, liarData$Creativity, alternative="less", method="kendall") • z = -3.2252, p-value = 0.0006294 • alternative hypothesis: true tau is less than 0 • sample estimates: • tau 47• -0.3002413 Reporting Kendall’s test A Kendall tau correlation coefficient indicated a correlation between creativity and performance in the World’s biggest liar contest, t=-.30, p<0.001 48 Biserial and point-biserial correlations • one variable is dichotomous (categorical with 2 categories) • point biserial: for discrete dichotomy (e.g., dead) • biserial: for continuous dichotomy (e.g., pass exam) 49 Readings • Discovering statistics using R (Andy Field, Jeremy Miles, Zoe Field) 50 R Title Text 51 set work directory • setwd("/new/work/directory") • getwd() • ls() # list the objects in the current workspace 52 packages • install.packages(“package.name") #installing packages • library(package.name) # loading a package • package::function() # disambiguating functions 53 Nominal and Ordinal data • mydata$v1 <- factor(mydata$v1, levels = c(1,2,3), labels = c("red", "blue", “green")) • mydata$v1 <- ordered(mydata$y, levels = c(1,3, 5), labels = c("Low", "Medium", "High")) 54 Missing data • is.na(var) #tests for missing valua/ also in rows • mydata$v1[mydata$v1==99] <- NA # select rows where v1 is 99 and recode column v1 • x <- c(1,2,NA,3) mean(x) # returns NA mean(x, na.rm=TRUE) • newdata <- na.omit(mydata) # spawn dataset without 55 missing data Install and load packages • install.packages(“car”); install.packages(“ggplot2”); install.packages(“pastecs”); install.packages(“psych”); install.packages(“descr”) • library(car);library(ggplot2);library(pastecs);librar y(psych);library(Rcmdr);library(descr) 56 Enter data • id<-c(1,2,3,4,5,6,7,8,9,10) • sex<-c(1,1,1,1,1,2,2,2,2,2) • courses<-c(2.0,1.0,1.0,2.0,3.0,3.0,3.0,2.0,4.0,3.0) • sex<-factor(sex, levels=c(1:2), labels=c("M", "F")) • example<- data.frame(ID=id,Gender=sex,Courses=courses ) 57 Frequency Distributions • facebook<- c(22,40,53,57,93,98,103,108,116,121,252) • library(modeest) • mfv(facebook) • mean(facebook) 58• median(facebook) Enter data • quantile (facebook) • IQR (facebook) • var(facebook) • sd(facebook) 59 describing your data • #load meaningful data • lecturer<-read.csv(“lecturerData.csv”, header=TRUE) • #get statistics • stat.desc(lecturerData[,c("friends", "income")], basic=FALSE, norm=TRUE) 60 describing your data II • # print frequency table • tab.friends<- as.data.frame(freq(ordered(lecturerData $friends)), plot=FALSE) • tab.friends.cumsum<-cumsum(tab.friends[- dim(tab.friends)[1],]$Frequency) • tab.friends$CumFreq=c(tab.friends.cumsum,NA) • tab.friends 61 Testing Normally Distributed • Load DLF data • dlf<-read.delim("DownloadFestival.dat", header=TRUE) • Data