Data Analysis Using R:2. Descriptive Statistics

Tuan V. Nguyen

Garvan Institute of Medical Research,

Sydney, Australia

Overview

• Measurements• Population vs sample• Summary of data: mean, variance, standard deviation,

standard error• Graphical analyses• Transformation

Scales of Measurement

• In general, most observable behaviors can be measured on a ratio-scale

• In general, many unobservable psychological qualities (e.g., extraversion), are measured on interval scales

• We will mostly concern ourselves with the simple categorical (nominal) versus continuous distinction (ordinal, interval, ratio)

categorical continuous

ordinal

interval

ratio

variables

Ordinal Measurement

• Ordinal: Designates an ordering; quasi-ranking– Does not assume that the intervals between numbers are equal.– finishing place in a race (first place, second place)

1 hour 2 hours 3 hours 4 hours 5 hours 6 hours 7 hours 8 hours

1st place 2nd place 3rd place 4th place

Interval and Ratio Measurement

• Interval: designates an equal-interval ordering– The distance between, for example, a 1 and a 2 is the

same as the distance between a 4 and a 5– Example: Common IQ tests are assumed to use an

interval metric

• Ratio: designates an equal-interval ordering with a true zero point (i.e., the zero implies an absence of the thing being measured)– Example: number of intimate relationships a person has

had• 0 quite literally means none• a person who has had 4 relationships has had twice as many

as someone who has had 2

Statististics: Enquiry to the unknown

Population Sample

Parameter Estimate

Estimate the population mean

Population height mean = 160 cm

Standard deviation = 5.0 cm

ht <- rnorm(10, mean=160, sd=5)mean(ht)

ht <- rnorm(10, mean=160, sd=5)mean(ht)

ht <- rnorm(100, mean=160, sd=5)mean(ht)

ht <- rnorm(1000, mean=160, sd=5)mean(ht)

ht <- rnorm(10000, mean=160, sd=5)mean(ht)hist(ht)

The larger the sample, the more accurate the estimate is!

Estimate the population proportion

Population proportion of males = 0.50 Take n samples, record the number of k males

rbinom(n, k, prob)

males <- rbinom(10, 10, 0.5)malesmean(males)

males <- rbinom(20, 100, 0.5)malesmean(males)

males <- rbinom(1000, 100, 0.5)malesmean(males)

The larger the sample, the more accurate the estimate is!

Summary of Continuous Data

• Measures of central tendency:– Mean, median, mode

• Measures of dispersion or variability:– Variance, standard deviation, standard error– Interquartile range

R commandslength(x), mean(x), median(x), var(x), sd(x)

summary(x)

R example

height <- rnorm(1000, mean=55, sd=8.2)mean(height)[1] 55.30948

median(height)[1] 55.018

var(height)[1] 68.02786

sd(height)[1] 8.2479

summary(height) Min. 1st Qu. Median Mean 3rd Qu. Max. 28.34 49.97 55.02 55.31 60.78 85.05

Graphical Summary: Box plot3

04

05

06

07

08

0boxplot(height)

95% percentile

75% percentile

25% percentile

5% percentile

Median, 50% perc.

Strip chart

30 40 50 60 70 80

Histogram

Histogram of height

height

Fre

qu

en

cy

30 40 50 60 70 80 90

05

01

00

15

02

00

25

0

Implications of the mean and SD

• “In the Vietnamese population aged 30+ years, the average of weight was 55.0 kg, with the SD being 8.2 kg.”

• What does this mean?

• 68% individuals will have height between 55 +/- 8.2*1 = 46.8 to 63.2 kg

• 95% individuals will have height between 55 +/- 8.2*1.96 = 38.9 to 71.1 kg

Implications of the mean and SD

• The distribution of weight of the entire population can be shown to be:

0

1

2

3

4

5

6

22 25 28 31 34 37 40 43 46 49 52 55 58 61 64 67 70 73 76 79 82 85 88 92

Weight (kg)

Per

cen

t (%

)

1SD

1.96SD

Summary of Categorical Data

• Categorical data: – Gender: male, female

– Race: Asian, Caucasian, African

• Semi-quantitative data: – Severity of disease: mild, moderate, severe

– Stages of cancer: I, II, III, IV

– Preference: dislike very much, dislike, equivocal, like, like very much

Mean and variance of a proportion

• For an individual i consumer, the probability he/she prefers A is pi. Assuming that all consumers are independent, then pi = p.

• Variance of pi is var(pi) = p(1-p)

• For a sample of n consumers, the estimated probability of preference for A is:

n

ppppp n

...321

and the variance of p_bar is:

n

ppp

1var

Normal approximation of a binomial distribution

• For an individual i consumer, the probability he/she prefers A is pi. Assuming that all consumers are independent, then pi = p.

• Variance of pi is var(pi) = p(1-p)

• For a sample of n consumers, the estimated probability of preference for A is:

n

ppppp n

...321

and the variance of p_bar is:

n

ppp

1var

and standard deviation: n

pps

1

Normal approximation of a binomial distribution - example

• 10 consumbers, 8 preferred product A.

• Proportion of preference for A: p = 0.8

• Variance: var(p) = 0.8(0.2)/10 = 0.016

• Standard deviation of p: s = 0.126

• 95% CI of p: 0.8 + 1.96(0.126) = 0.55 to 1.00

Descriptive AnalysesContinuous data

Paired t-test

• Continuous data• Normally distributed• Two samples are NOT independent

Paired t-test – an example

• The problem: Viewing certain meats under red light might enhance judges preferences for meat. 12 judges were asked to score the redness of meat under red light and white light

Results:

Judge Red White

1 20 22

2 18 19

3 19 17

4 22 18

5 17 21

6 20 23

7 19 19

8 16 20

9 21 22

10 17 20

11 23 27

12 18 24

Paired t-test – analysis

Judge Red light White light Difference

1 20 22 2

2 18 19 1

3 19 17 -2

4 22 18 -4

5 17 21 4

6 20 23 3

7 19 19 0

8 16 20 4

9 21 22 1

10 17 20 3

11 23 27 4

12 18 24 6

Mean 21.0 19.2 1.83

SD 2.8 2.1 2.82

Mean difference: 1.83, SD: 0.81

Standard error (SE):

SD/sqrt(n) = 0.81/sqrt(10) = 0.81

T-test = (1.83 – 0)/0.81 = 2.23

P-value = 0.0459

Conclusion: there was a significant effect of light colour.

Paired t-test – R analysis

red < -c(20,18,19,22,17,20,19,16,21,17,23,18)

white < -c(22,19,17,18,21,23,19,20,22,20,27,24)

t.test(red, white, paired=TRUE)

data: red and white t = -2.2496, df = 11, p-value = 0.04592alternative hypothesis: true difference in means is not

equal to 0 95 percent confidence interval: -3.6270234 -0.0396433 sample estimates:mean of the differences -1.833333

Two-sample t-test

Sample Group 1 Group21 x1 y1

2 x2 y2 3 x3 y3 4 x4 y4 5 x5 y5 … …n xn yn

Sample size n1 n2

Mean x y

SD sx sy

Mean difference:

D = x – y

Variance of D:

T-statistic:

95% Confidence interval:

Two-group comparison: an example

ID A B

1 3 3

2 7 1

3 1 2

4 9 4

5 3 5

6 4 2

7 1 2

8 2 5

9 6 3

10 7 2

ID AB

11 5 3

12 8 4

13 5 2

14 9 3

15 4 5

16 6 4

17 4 3

18 3 1

19 9 3

20 5 2

20 consumers rated their preference for two rice desserts (A and B)

Unpaired t-test using R

a<-c(3,7,1,9,3,4,1,2,6,7,5,8,5,9,4,6,4,3,9,5)b<-c(3,1,2,4,5,2,2,5,3,2,3,4,2,3,5,4,3,1,3,2)t.test(red,white)

Welch Two Sample t-test

data: a and b

t = 3.3215, df = 27.478, p-value = 0.002539

alternative hypothesis: true difference in means is not equal to 0

95 percent confidence interval:

0.8037895 3.3962105

sample estimates:

mean of x mean of y

5.05 2.95

Transformation of data: multiplicative effects

• The following data represent lysozyme levels in the gastric juice of 29 patients with peptic ulcer and of 30 normal controls. It was interested to know whether lysozyme levels were different between two groups.

Group 1:

0.2 0.3 0.4 1.1 2.0 2.1 3.3 3.8 4.5 4.8 4.9 5.0 5.3 7.5 9.8 10.4 10.9 11.3 12.4 16.2 17.6 18.9 20.7 24.0 25.4 40.0 42.2 50.0 60.0

Group 2:

0.2 0.3 0.4 0.7 1.2 1.5 1.5 1.9 2.0 2.4 2.5 2.8 3.6 4.8 4.8 5.4 5.7 5.8 7.5 8.7 8.8 9.1 10.3 15.6 16.1 16.5 16.7 20.0 20.7 33.0

Unpaired t-test by Rg1 <- c( 0.2, 0.3, 0.4, 1.1, 2.0, 2.1, 3.3, 3.8,

4.5, 4.8, 4.9, 5.0, 5.3, 7.5, 9.8, 10.4,

10.9, 11.3, 12.4, 16.2, 17.6, 18.9, 20.7,

24.0, 25.4, 40.0, 42.2, 50.0, 60)

g2 <- c(0.2, 0.3, 0.4, 0.7, 1.2, 1.5, 1.5, 1.9, 2.0,

2.4, 2.5, 2.8, 3.6, 4.8, 4.8, 5.4, 5.7, 5.8,

7.5, 8.7, 8.8, 9.1, 10.3, 15.6, 16.1, 16.5,

16.7, 20.0, 20.7, 33.0)

t.test(g1, g2)

data: g1 and g2 t = 2.0357, df = 40.804, p-value = 0.04831alternative hypothesis: true difference in means is not equal to 0 95 percent confidence interval: 0.05163216 13.20239083 sample estimates:mean of x mean of y 14.310345 7.683333

Exploration of data

par(mfrow=c(1,2))

hist(g1)

hist(g2)

Histogram of g1

g1

Fre

qu

en

cy

0 10 20 30 40 50 60

05

10

15

Histogram of g2

g2

Fre

qu

en

cy

0 5 10 20 30

05

10

15

Group 1:

mean(g1) = 14.3

sd(g1) = 15.7

Group 2:

mean(g2) = 7.7

sd(g2) = 7.8

Re-analysis of lysozyme data

log.g1 <- log(g1)

log.g2 <- log(g2)

t.test(log.g1, log.g2)

data: log.g1 and log.g2 t = 1.406, df = 55.714, p-value = 0.1653alternative hypothesis: true difference in means is not equal to 0 95 percent confidence interval: -0.2182472 1.2453165 sample estimates:mean of x mean of y 1.921094 1.407559

exp(1.921-1.407) = 1.67

Group 1’s mean is 67% higher than group 2’s

Descriptive analysisCategorical data

Comparison of two proportions - theory

Group1 2

____________________________________________

Sample size n1 n2

Number of events e1 e2

Proportion of events p1 p2

Difference: D = p1 – p2 SE difference: SE = [p1(1–p1)/n1 + p2(1–p2)/n2]1/2

Z = D / SE95% CI: D + 1.96(SE)

With (n1 + n2) > 20, and if Z > 2, it is possible to reject the null hypothesis.

Comparison of two proportions - example

GroupHeroine Cocaine

__________________________________________

Sample size 100 100Number of deaths 90 36Mortality rate 0.90 0.36

Thirty-day mortality rate (%) of 100 rats who had been exposed to heroine or cocain.

Analysis

Difference: D = 0.90 – 0.36 = 0.54SE (D) = [0.9(0.1)/100 + 0.36(0.64)/100]1/2

= 0.057Z = 0.54 / 0.057 = 9.54

95% CI:0.54 + 1.96(0.057)0.43 to 0.65

Conclusion: reject the null hypothesis.

Comparison of two proportions - R

events <- c(90, 36)

total <- c(100, 100)

prop.test(events, total)

2-sample test for equality of proportions with continuity correction

data: deaths out of total X-squared = 60.2531, df = 1, p-value = 8.341e-15alternative hypothesis: two.sided 95 percent confidence interval: 0.4190584 0.6609416 sample estimates:prop 1 prop 2 0.90 0.36

Comparison of >2 proportions – Chi square analysis

table(sex, ethnicity)

ethnicity

sex African Asian Caucasian Others

Female 4 43 22 0

Male 4 17 8 2

females <- c(4, 43, 22, 0)

total <- c(8, 60, 30, 2)

prop.test(females, total)

Comparison of >2 proportions – Chi square analysis

4-sample test for equality of proportions without continuity

correction

data: females out of total X-squared = 6.2646, df = 3, p-value = 0.09942alternative hypothesis: two.sided sample estimates: prop 1 prop 2 prop 3 prop 4 0.5000000 0.7166667 0.7333333 0.0000000

Warning message:Chi-squared approximation may be incorrect in:

prop.test(females, total)

Summary

• Examine the distribution of data– Mean and variance: systematic difference?

– Normally distributed ?

• Transformation?

• Present confidence intervals (and p-values)