Data Analysis Using R: 2. Descriptive Statistics

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Data Analysis Using R2. Descriptive Statistics

• Tuan V. Nguyen
• Garvan Institute of Medical Research,
• Sydney, Australia

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Overview

• Measurements
• Population vs sample
• Summary of data mean, variance, standard
  deviation, standard error
• Graphical analyses
• Transformation

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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)

variables
categorical
continuous
ordinal
interval
ratio
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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)

1st place
2nd place
3rd place
4th place
1 hour 2 hours 3 hours 4 hours 5 hours 6 hours 7
hours 8 hours
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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

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Statististics Enquiry to the unknown
Population Sample
Parameter Estimate
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Estimate the population mean

• Population height mean 160 cm
• Standard deviation 5.0 cm

ht lt- rnorm(10, mean160, sd5) mean(ht) ht lt-
rnorm(10, mean160, sd5) mean(ht) ht lt-
rnorm(100, mean160, sd5) mean(ht) ht lt-
rnorm(1000, mean160, sd5) mean(ht) ht lt-
rnorm(10000, mean160, sd5) mean(ht) hist(ht)
The larger the sample, the more accurate the
estimate is!
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Estimate the population proportion

• Population proportion of males 0.50
• Take n samples, record the number of k males
• rbinom(n, k, prob)

males lt- rbinom(10, 10, 0.5) males mean(males) ma
les lt- rbinom(20, 100, 0.5) males mean(males) mal
es lt- rbinom(1000, 100, 0.5) males mean(males)
The larger the sample, the more accurate the
estimate is!
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Summary of Continuous Data

• Measures of central tendency
• Mean, median, mode
• Measures of dispersion or variability
• Variance, standard deviation, standard error
• Interquartile range

R commands length(x), mean(x), median(x),
var(x), sd(x) summary(x)
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R example

• height lt- rnorm(1000, mean55, sd8.2)
• 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

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Graphical Summary Box plot
boxplot(height)
95 percentile
75 percentile
Median, 50 perc.
25 percentile
5 percentile
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Strip chart
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Histogram
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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 /-
  8.21 46.8 to 63.2 kg
• 95 individuals will have height between 55 /-
  8.21.96 38.9 to 71.1 kg

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Implications of the mean and SD

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

1.96SD
1SD
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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

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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

and the variance of p_bar is
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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

and the variance of p_bar is
and standard deviation
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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

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Descriptive AnalysesContinuous data
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Paired t-test

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

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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 1
0 17 20 11 23 27 12 18 24
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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.
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Paired t-test R analysis

• red lt -c(20,18,19,22,17,20,19,16,21,17,23,18)
• white lt -c(22,19,17,18,21,23,19,20,22,20,27,24)
• t.test(red, white, pairedTRUE)

data red and white t -2.2496, df 11,
p-value 0.04592 alternative 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
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Two-sample t-test
Mean difference D x y Variance of D
Sample Group 1 Group2 1 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
T-statistic
95 Confidence interval
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Two-group comparison an example
20 consumers rated their preference for two rice
desserts (A and B)

• 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 A B 11 5 3 12 8 4 13 5 2 14 9 3 15 4 5 16 6 4 1
7 4 3 18 3 1 19 9 3 20 5 2
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Unpaired t-test using R

• alt-c(3,7,1,9,3,4,1,2,6,7,5,8,5,9,4,6,4,3,9,5)
• blt-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 alternativ
e 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
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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

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Unpaired t-test by R

• g1 lt- 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 lt- 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 0.04831 alternative 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
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Exploration of data

• par(mfrowc(1,2))
• hist(g1)
• hist(g2)

Group 1 mean(g1) 14.3 sd(g1) 15.7 Group
2 mean(g2) 7.7 sd(g2) 7.8
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Re-analysis of lysozyme data

• log.g1 lt- log(g1)
• log.g2 lt- log(g2)
• t.test(log.g1, log.g2)

data log.g1 and log.g2 t 1.406, df 55.714,
p-value 0.1653 alternative 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 1s mean is 67
higher than group 2s
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Descriptive analysisCategorical data
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Comparison of two proportions - theory
Group 1 2 ___________________
_________________________ Sample
size n1 n2 Number of events e1 e2 Proportion of
events p1 p2
Difference D p1 p2 SE difference SE
p1(1p1)/n1 p2(1p2)/n21/2 Z D / SE 95 CI
D 1.96(SE) With (n1 n2) gt 20, and if Z gt
2, it is possible to reject the null hypothesis.
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Comparison of two proportions - example
Thirty-day mortality rate () of 100 rats who had
been exposed to heroine or cocain.
Analysis Difference D 0.90 0.36 0.54 SE
(D) 0.9(0.1)/100 0.36(0.64)/1001/2
0.057 Z 0.54 / 0.057 9.54 95 CI 0.54
1.96(0.057) 0.43 to 0.65 Conclusion reject
the null hypothesis.
Group Heroine Cocaine __________________
________________________ Sample
size 100 100 Number of deaths 90 36 Mortality
rate 0.90 0.36
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Comparison of two proportions - R

• events lt- c(90, 36)
• total lt- 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-15 alternative hypothesis two.sided 95
percent confidence interval 0.4190584 0.6609416
sample estimates prop 1 prop 2 0.90 0.36
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Comparison of gt2 proportions Chi square
analysis

• table(sex, ethnicity)
• ethnicity
• sex African Asian Caucasian Others
• Female 4 43 22 0
• Male 4 17 8 2

females lt- c(4, 43, 22, 0) total lt- c(8, 60, 30,
2) prop.test(females, total)
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Comparison of gt2 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 0.09942
• alternative 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)

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Summary

• Examine the distribution of data
• Mean and variance systematic difference?
• Normally distributed ?
• Transformation?
• Present confidence intervals (and p-values)