Descriptive Methods

707.031: Evaluation Methodology Winter 2015/16

Eduardo Veas

what we do with the data depends on the scales

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Measurement Scales

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The complexity of measurements

• Nominal

• Ordinal

• Interval

• Ratio

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Sophisticated

Crude

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

• use: freq, counts, 5

Ordinal data

• ranking of an attribute

• interval between points in scale not intrinsically equal

• comparisons < or > are possible

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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)

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Frequencies

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Title Text

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

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Interpreting frequency tables

Frequency Percent CumPercent CumFreq1 2 20 20 22 3 30 50 53 4 40 90 94 1 10 100 10Total 10 100 NA NA

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Contingency Tables

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Right-handed Left-handed Total

Males 43 9 52

Females 44 4 48

Totals 87 13 100

sd

Modelling

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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

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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

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Histogram / Frequency distributions

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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• can be influenced by extreme scores17

Dispersion of a distribution

• range: difference between lowest and highest score

• interquartile difference: mode + upper and lower quartiles

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252 - 22 = 232 121 - 22 = 99

Fit of the mean

• deviance: mean - x

• sum of squared errors (SS)

• variance = SS / N-1

• stddev = sqrt(variance)19

Assumptions

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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 influence that of another21

Distributions for DLF

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0 1 2 3 4Hygiene score on day1

Density

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0 1 2 3Hygiene score on day 2

Density

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Density

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sample

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Quantify normallity

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Different groups

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Exam histogram

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25 50 75 100exam

density

Exam histogram

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density

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60 70 80 90 100exam

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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)

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Reporting Shapiro-Wilk

• A Shapiro-Wilk test on the R exam, W=0.96, proved a significant deviation from normality (p<0.05).

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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

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Homogeneity of variance

• Levene in large datasets may give sig for small variations

• Double check Variance ratio (Hartley’s Fmax)

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Correlations

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Title Text

Everything is hard to begin with, but the more you practise the easier it gets

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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 skill33

Correlations

• Bivariate: correlation between two variables

• Partial: correlation between two variables while controlling the effect of one or more additional variables

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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)

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Covariance II

• Positive: both variables vary in the same direction

• Negative: variables vary in opposite directions

• Covariance is scale dependent and cannot be generalized

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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

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Dataset Exams and Anxiety

• effects of exam stress and revision on exam performance

• questionnaire to assess anxiety relating to exams (EAQ)

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Enter data

• examData<-read.delim("ExamAnxiety.dat", header=TRUE)

• examData2<-examData[,c(“Exam”,"Anxiety","Revise")]

• cor(examData2)

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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

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Confidence values

• rcorr(as.matrix(examData[,c(“Exam","Anxiety","Revise")]))

• Exam Anxiety Revise• Exam 0 0 • Anxiety 0 0 • Revise 0 0

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Reporting Pearson’s CC

A Pearson correlation coefficient indicated a significant correlation between anxiety performance and time spent revising, r=-.44, p<0.01

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Spearman’s correlation coefficient

• non parametric test

• first rank the data and then apply Pearson cc

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Liar Dataset

• contest for storytelling the biggest lie

• 68 participants, ranking, and creativity questionnaire

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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

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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

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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 • -0.3002413 47

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

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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)

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Readings

• Discovering statistics using R (Andy Field, Jeremy Miles, Zoe Field)

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R

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Title Text

set work directory

• setwd("/new/work/directory")

• getwd()

• ls()    # list the objects in the current workspace

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packages

• install.packages(“package.name") #installing packages

• library(package.name) # loading a package

• package::function() # disambiguating functions

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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"))

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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 missing data55

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);library(psych);library(Rcmdr);library(descr)

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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)

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Frequency Distributions

• facebook<-c(22,40,53,57,93,98,103,108,116,121,252)

• library(modeest)• mfv(facebook)

• mean(facebook)

• median(facebook)58

Enter data

• quantile (facebook)

• IQR (facebook)

• var(facebook)

• sd(facebook)

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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)

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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

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Testing Normally Distributed

• Load DLF data• dlf<-read.delim("DownloadFestival.dat",

header=TRUE)

• Data about hygiene collected during a festival (3days)

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Enter data

• hist.day1 <- ggplot (dlf, aes(day1)) + theme(legend.position = "none") + geom_histogram(aes(y = ..density..), colour="black", fill="white")+ labs(x="Hygiene score on day1", y="Density")

• hist.day1 + stat_function(fun = dnorm, args=list(mean=mean(dlf$day1,na.rm=TRUE), sd=sd(dlf$day1, na.rm = TRUE)), colour="blue", size=1)

• qqplot.day1 <-qplot(sample=dlf$day1, stat="qq")63

Plot day 1

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0.0

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0 5 10 15 20Hygiene score on day1

Density

Offending score

• # print bad score• dlf$day1[dlf$day1>5]

• #correct bad score• dlf$day1[dlf$day1>5]<-2.02

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Quantify normallity

• describe(cbind(dlf$day1, dlf$day2, dlf$day3))

• stat.desc(dlf[,c("day1","day2","day3")p], basic = FALSE, norm= TRUE)

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Groups

• rexam<-read.delim("rexam.dat", header=TRUE)

• # obtain statistics for exam, computer, lectures and numeracy• round(stat.desc(rexam[,c("exam","computer","lectures","numer

acy")], basic=FALSE, norm=TRUE), digits=3)

• hist.exam <-ggplot (rexam, aes(exam)) + theme(legend.position = "none") + geom_histogram(aes(y = ..density..), colour="black", fill="white") + labs(x="exam", y="density") + stat_function(fun=dnorm, args=list(mean=mean(rexam$exam,na.rm=TRUE), sd=sd(rexam$exam, na.rm=TRUE)), colour="blue", size=1)67

Add factors

• # set uni to be a factor• rexam$uni <-factor(rexam$uni, levels = c(0:1),

labels = c("KFU", “TUG"))

• by (rexam[, c("exam", "computer", "lectures", "numeracy")], rexam$uni, stat.desc, basic=FALSE, norm = TRUE)

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Get subsets and individual histograms

• # now we create subsets of the example datasets for each factor

• kfu<-subset(rexam, rexam$uni=="KFU")• tug<- subset(rexam, rexam$uni==“TUG")

• # now we can create histograms for each subset• hist.exam.kfu <-ggplot (kfu, aes(exam)) +

theme(legend.position = "none") + geom_histogram(aes(y = ..density..), colour="black", fill="white") + labs(x="exam", y="density") + stat_function(fun=dnorm, args=list(mean=mean(kfu$exam,na.rm=TRUE), sd=sd(kfu$exam, na.rm=TRUE)), colour="blue", size=1)69