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Data Handling II: Describing and Depicting your Data Dr Yanzhong Wang Lecturer in Medical Statistics Division of Health and Social Care Research King's College London Email: yanzhong.wang@kcl.ac.ukyanzhong.wang@kcl.ac.uk Drug Development Statistics & Data Management

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2 Types of data Quantitative data – continuous, discrete – distributions may symmetric or skewed Qualitative (categorical) data – binary – nominal, ordinal

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3 Long tail to leftLong tail to right Skewed Distributions

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4 0246 0.1.2.3.4 Symmetric Distribution

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5 Summary statistics ‘Where the data are’ - location – mean, median, mode, geometric mean Used to describe baseline data and main outcomes ‘How variable the data are’ - spread – standard deviation, variance, range, interquartile range, 95% range Needed (primarily) to describe baseline data in RCT and cohort study

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6 Definition of the Mean The mean of a sample of values is the arithmetic average and is determined by dividing the sum of the values by the number of the values.

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7 Definition of the Median The median is the middle value. not affected by skewness and outliers, but less precise than mean theoretically.

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Ordered Blood Glucose Values 2.2 2.9 3.3 3.3 3.3 3.4 3.4 3.4 3.6 3.6 3.6 3.6 3.7 3.7 3.8 3.8 3.8 3.9 4.0 4.0 4.0 4.1 4.1 4.1 4.2 4.3 4.4 4.4 4.4 4.5 4.6 4.7 4.7 4.7 4.8 4.9 4.9 5.0 5.1 6.0 8

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Definition of the Mode The mode is the most frequent value. 9

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2.2 2.9 3.3 3.3 3.3 3.4 3.4 3.4 3.6 3.6 3.6 3.6 3.7 3.7 3.8 3.8 3.8 3.9 4.0 4.0 4.0 4.1 4.1 4.1 4.2 4.3 4.4 4.4 4.4 4.5 4.6 4.7 4.7 4.7 4.8 4.9 4.9 5.0 5.1 6.0 Ordered Blood Glucose Values 10

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0 1 2 3 4 5 6 7 2 3 4 5 6 Blood glucose (mmol/litre) Count Arithmetic Mean - outlier prone Mode - not necessarily central (categorical data) Median - only uses relative magnitudes Location = Central Tendency 11

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Relation of mean, median and mode If distribution is unimodal (has only one mode) then: Mean=median=mode for symmetric distribution. Mean>median>mode for positively skewed distribution. Mean

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0 10 20 30 40 50 60 70 80 0.20.30.40.50.60.70.80.911.11.21.31.41.51.61.7 Serum Triglyceride Levels Count Serum Triglyceride Levels from Cord Blood of 282 Babies 13

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0 5 10 15 20 25 30 35 -1.9-1.7-1.5-1.3-1.1-0.9-0.7-0.5-0.3-0.10.10.30.5 log(Serum Triglyceride) Levels count Log(Serum Triglyceride Levels) from Cord Blood of 282 Babies 14

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Definition of the Geometric Mean The geometric mean of a sample of n values is determined by multiplying all the values together and taking the nth root (for only two values this is the more familiar square root). 15

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Geometric Mean A common example of when the geometric mean is the correct choice average is when averaging growth rates. Another Method: Take log of each value, find arithmetic mean and anti-log the result. Exp( (log(0.15) + … + log(1.66) )/40) = 0.467

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0 10 20 30 40 50 60 70 80 0.20.30.40.50.60.70.80.911.11.21.31.41.51.61.7 Serum Triglyceride Levels Count Mean=0.506 Median=0.460 Geometric Mean=0.467 Serum Triglyceride Levels from Cord Blood of 282 Babies 17

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Why measures of variability are important Production of Aspirin New production process of 100 mg tabs Random sample from process – 96 97 100 101 101 mgs - mean 99 mg Random sample from old process – 88 93 100 104 110 mgs - mean 99 mg Same means but new is better because less variable 18

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Definition of Range The range of a sample of values is the largest value minus the smallest value. New process the range is 101-96=5 Old process the range is 110-88=22 Range is simple ….. BUT – Only uses min and max – Gets larger as sample size increases 19

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Definition of Inter-quartile Range The inter-quartile range of a sample of values is the difference between the upper and lower quartiles. The lower quartile is the value which is greater than ¼ of the sample and less than ¾ of the sample. Conversely, the upper quartile is the value which is greater than ¾ of the sample and less than ¼ of the sample. 20

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Ordered Blood Glucose Values 2.2 2.9 3.3 3.3 3.3 3.4 3.4 3.4 3.6 3.6 3.6 3.6 3.7 3.7 3.8 3.8 3.8 3.9 4.0 4.0 4.0 4.1 4.1 4.1 4.2 4.3 4.4 4.4 4.4 4.5 4.6 4.7 4.7 4.7 4.8 4.9 4.9 5.0 5.1 6.0 1/4 of 40 = 10 3/4 of 40 = 30 21

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0 1 2 3 4 5 6 7 2 3 4 5 6 Blood glucose (mmol/litre) Count Inter-Quartile Range Lower quartile Upper quartile Inter-quartile range 22

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Standard deviation Neither measure uses the numerical values - only relative magnitudes A measure accounting for the values is the standard deviation Consider the aspirin data from the new process 96 97 100 101 101 (mean 99 mg) Determine deviations from mean -3 -2 1 2 2 Square, add, average and square-root 23

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Measures of scatter/dispersion – ‘how variable the data are’ Range – smallest to biggest value – increases with sample size Standard deviation – measure of variation around the mean – affected by skewness and outliers Variance = square of standard deviation Interquartile range (IQR) – from 25th centile to 75th centile 24

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Plotting Data Histograms Stem and Leaf Plots Box Plots Stem Leaf 60 0 1 58 56 54 52 50 00 2 48 000 3 46 0000 4 44 0000 4 42 00 2 40 000000 6 38 0000 4 36 000000 6 34 000 3 32 000 3 30 28 0 1 26 24 22 0 1 ----+----+----+----+ Multiply Stem.Leaf by 10**-1 2 3 4 5 6 Blood glucose (mmol/litre) 25

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Mean and standard deviation Best description if distribution reasonably symmetric (and single mode) Give full description if data have Normal distribution 26

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Mean 3, s.d. 1 Mean 5, s.d. 1 Mean 5, s.d. 2 27

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Properties of Normal distribution Symmetric distribution – mean, median and mode equal Completely specified by mean and standard deviation 95% of distribution contained within mean 1.96 standard deviations 68% within mean 1 standard deviation 28

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Continuous data, not Normally distributed If symmetric use mean and standard deviation If skewed use median and IQR Unless Positively skewed, but log transformation creates symmetric distribution – use geometric mean 29

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Nominal categorical data Mode. % in each category, especially when binary. Wheeze in last 12 months Frequency (n)% No194575.2 Yes64224.8 Total2587100.0 30

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Ordinal categorical data Median and IQR if enough separate values. Otherwise as for nominal. 31

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Discrete quantitative data As for continuous data if many values, as for ordinal data if fewer.

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33 Difference Between Standard Deviation & Standard Error

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34 Measure of Variability of the Sample Mean Range, inter-quartile range and standard deviation relate to population (sample) not mean. To understand the difference carry out a sampling experiment using the Ritchie Index values

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35 Values of the Ritchie Index (Measure of Joint Stiffness) in 50 Untreated Patients 14 9 8 9 1 20 3 3 2 4 2 3 6 1 2 11 16 24 16 21 19 22 33 12 12 12 19 10 33 2 19 40 1 20 1 2 4 7 9 4 9 6 14 8 27 10 27 7 24 21 Mean = (14+…+21)/50 = 12.18

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36 0 2 4 6 8 10 12 14 16 0 - 56 - 1011 - 1516 - 2021 - 2526 - 3031 - 3536 - 40 Values of the Ritchie Index Arithmetic Mean - outlier prone Median - only uses relative magnitudes Mode - not necessarily central (categgorical data) Location = Central Tendency

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37 Sampling Experiment Take a random sample (10) from the 50 values Determine the mean of the 10 values Repeat 50 times These means show variation - HOW LARGE IS IT ?

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38 Variations in Samples 0 2 4 6 8 10 12 14 16 0 - 56 - 1011 - 1516 - 2021 - 2526 - 3031 - 3536 - 40 Values of the Ritchie Index 0 2 4 6 8 10 12 14 16 0 - 56 - 1011 - 1516 - 2021 - 2526 - 3031 - 3536 - 40 Values of the Ritchie Index 0 2 4 6 8 10 12 14 16 0 - 56 - 1011 - 1516 - 2021 - 2526 - 3031 - 3536 - 40 Values of the Ritchie Index 0 2 4 6 8 10 12 14 16 0 - 56 - 1011 - 1516 - 2021 - 2526 - 3031 - 3536 - 40 Values of the Ritchie Index 0 2 4 6 8 10 12 14 16 0 - 56 - 1011 - 1516 - 2021 - 2526 - 3031 - 3536 - 40 Values of the Ritchie Index Mean=12.18 Mean=10.00 Mean=12.60 Mean=13.40 Mean=11.50

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39 Ritchie Values Values of the Ritchie Index 0 - 56 - 1011 - 1516 - 2021 - 2526 - 3031 - 3536 - 40 30 25 20 15 10 5 0 Original values (mean - 12.18 ; sd - 9.69)

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40 Ritchie Values Sampling Experiment – Sample Means Values of the Ritchie Index 0 - 56 - 1011 - 1516 - 2021 - 2526 - 3031 - 3536 - 40 30 25 20 15 10 5 0 Sample means (mean - 12.21 ; sd - 2.97) Original values (mean - 12.18 ; sd - 9.69)

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41 Definition of the Standard Error The standard deviation of the sampling distribution of the mean is called the standard error of the mean.

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42 Increasing Sample Size Increased precision (smaller standard error) Less skewness Values of the Ritchie Index 0 - 56 - 1011 - 1516 - 2021 - 2526 - 3031 - 3536 - 40 Sample means (mean - 12.21 ; sd - 2.97) 30 25 20 15 10 5 0 35 40 30 25 20 15 10 Values of the Ritchie Index 0 - 56 - 1011 - 1516 - 2021 - 2526 - 3031 - 3536 - 40 Sample means (mean - 12.37 ; sd - 2.43) 5 0 35 40 n=10 n=15

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43 Standard error of the mean as a function of the sample size 0 1 2 3 4 5 6 7 8 9 10 0 203040 Sample Size Standard Error of the Mean

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44 Population of Gene Lengths n=20,290 0 5000 10000 15000 Gene Length (# of nucleotides) Frequency 0 500 1000 1500 2000 2500 3000

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45 Samples of size : n=100 0 5000 10000 15000 0 50 100 150 200 250 300 Gene Length (# of nucleotides) Frequency

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46 Practical Confusion A mean is often reported in medical papers as 12.18 1.37 what is 1.37 ? sd or se ?

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Thanks! Tea break

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