1. Descriptive Statistics F. Farrokhyar, MPhil, PhD, PDoc Department of Surgery Department of Clinical Epidemiology and Biostatistics March 18, 2009

2. Objectives  To understand and recognize different types of variables  To learn how to explore your data ◙How to display data with numbers and tables ◙How to display data using graphs  To understand the fundamental concept of variability  To learn the notion of the distribution of a variable

3. Why and how are statistics relevant to medicine?  Prevention – What causes a disease?  Diagnosis – What symptoms and signs do patients with a given disease present with?  Treatment – What treatments are effective for a given disease and for which patients?  Prognosis – How will specific patients with a given disease fare in the long term?

4. Statistics – Why do we need it?

5. B A E W D S A Q P B B W E O N F O H E E R D T TY E D T E Q O N E G G O L T S D G F E W G E G G V B A Y A O E E D Y H E J U E G D E T E W W E T H E F E O P L U M R HOW MANY ‘ E”’s ?

6. Descriptive and Inferential statistics?  Descriptive statistics are concerned with the presentation, organization, and summarization of data  Inferential statistics allow us the generalization from a sample to a larger group of subjects.

7. What is data?  Data is collected for some purpose and each collected information have a meaning in some context.  Data is a set of information or observation about a group of individuals or subjects.  This information is organized in form of variables.  A variable is any characteristic of a person or a subject that can be measured or categorized and its value varies from individual to individual.

8. Dependent and Independent Variables? Independent variable  Is the explanatory variable that explains the changes in the dependent variable demographics (age, gender, height), risk factors (diabetes, CAD)  Is the intervention or exposure that causes the changes in the dependent variable. drug, surgery, radiation, smoking … Dependent variable  Is the outcome of interest, which changes in response to some intervention or exposure. mortality, survival, post-op pain, quality of life, post-op complications

9. Type of variables …? Qualitative or attribute variable  Nonnumeric gender, severity of injury, type of injury, tumour grade Categorical variables… Quantitative variable  Numeric Discrete variable can assume only whole numbers: number of accidents, number of injuries, pain score Continuous variable may take any value, within a defined range: weight, height, age, blood pressure, level of cholesterol, pain score

10. Level of measurement …  There are four level of measurement: ◙Nominal ◙Ordinal ◙Interval ◙Ratio Qualitative/Categorical Quantitative/Numeric

11. Level of measurement … cont’d  Variable type: ◙Nominal ◙Ordinal. ◙Interval. ◙Ratio  Assumptions: ◙Named categories ◙Same as nominal plus ordered categories ◙Same as ordinal plus equal intervals ◙Same as interval plus meaningful zero

12. Level of measurement … cont’d  A nominal variable: consists of named categories, with no implied order among the categories. - gender, mortality ---- dichotomous or binary - type of injury, type of fracture, blood type  An ordinal variable: consists of ordered categories, where the differences between categories cannot be considered to be equal. - Tumour stage – I, II, III, IV, tumour grade – I II, III, IV - Likert scale – excellent, very good, good, fair, poor

13. Level of measurement … cont’d  An interval variable: has equal distances between values with no meaningful ‘zero’ value. - IQ test (the differences between numbers are meaningful but the ratios between them are not)  An ratio variable: has equal intervals between values and a meaningful zero point. The ratio between them makes sense. - height, weight, laboratory test values, age

14. Primary objective: To compare the post-operative pain between laparoscopic and open surgery in patients with colorectal cancer Secondary objective: To compare the post-operative complications between laparoscopic and open surgery in patients with colorectal cancer For example

15. Independent (Explanatory) variables: Age, Sex, Pre-op pain Severity Dependent/outcome variables: Changes in pain, Complication Independent (Comparison) variable

16. Data Editing  Validity edits: Ensure that: essential fields have been completed and there are no missing information ◘specified units of measure have been properly used and the measurements are within the acceptable range.  Duplication edits: Ensure that each case/patient have been entered into the database only once.  Statistical edits: Identify and double check all the extreme values, suspicious data and outliers.

17. Descriptive Statistics  … are a means of organizing and summarizing observations.  We examine variables in order to describe their main features.  It is the basic strategies that help us organize our exploration of a set of data: ◙Begin by examining each variable. ◙Examine the distribution of each variable by creating frequency tables, numerical summaries and graphs. ◙Study the relationships between the variables.

18. Examining Distributions: Categorical …  Numbers Frequencies (counts), cumulative frequencies Relative frequencies (%), cumulative relative frequencies (%)  Graphs Bar charts Pie charts

19. Cross-tabulation of categorical data

21. Examining Distributions: Categorical … Numbers Frequencies (counts), cumulative frequencies Relative frequencies (%), cumulative relative frequencies (%)  Graphs Bar charts Pie charts

22. Bar Charts

24. Bar charts …  A bar chart can be used to depict any levels of measurement ( nominal, ordinal, interval, or ratio).  A series of separated bars (vertical or Horizontal), one per category.  Bars represent frequency (counts) or relative frequency (percent or proportion) of each category.  A Bar chart is also useful for showing data for more than one group.

25. Pie Charts

26. Pie charts …  Used primarily for nominal and ordinal data.  Used to display relative frequency distribution.  The circle is divided proportionally using relative frequency of each category.  A pie chart is useful for showing data for one group but it is useless for graphic illustration of two or more groups.

27. Examining Distributions: Quantitative …  Numbers Measures of central tendency – mean, median, mode Measures of variation around mean – variance, standard deviation, standard error of mean Measures of variation around median – percentiles, quintiles, quartiles  Graphs Histograms The five-number summary  Box plots

28.  Mean: sum of observations divided by number of observations  Median: is a midpoint of a distribution after arranging all observations in order of size, from smallest to largest.  Mode: most frequent value – the highest peak Measures of central tendency

29. Properties of mean …  It is used for interval or ratio data.  A set of data has only a mean.  All values are included in the computation.  It is the only measure of central tendency where the sum of deviations of each value from the mean will always be zero.  The mean is a useful measures for comparing two or more sets of data.  The mean is sensitive toward extreme values.

30. Properties of median …  It is used for interval or ratio data.  There is a unique median for each data set.  The median is not necessarily equal to one of the sample values.  It is resistant (insensitive) toward extreme values.  It is useful for summarising skewed data.

31.  Variance: the average of the squares of the deviations of the data from their mean  Standard deviation : square root of variance  Standard error : Measures of variation around mean

32. Properties of variance …  All values are used on calculation.  The units are not the same as data, they are the square of the original units.

33. Properties of standard deviation …  The units are the same as data  It is used for Empirical Rule.  For any symmetrical distribution: ◘About 68% of the observations will lie within 1 s. d. of the mean. ◘About 95% of the observations will lie within 2 s. d. of the mean. ◘About 99.8% of the observations will lie within 3 s. d. of the mean.

34. The Empirical Rule

35. Measures of variation around median  Percentiles: Arrange the observations from smallest to largest. Divide into 100 equal parts; for example; the 5 th percentiles of a distribution is the value which 5% of the observations fall below and 95% fall above.  Quartiles: 25 th, 50 th and 75 th percentiles  Quintiles: 20 th, 40 th, 60 th, and 80 th percentiles  Deciles: 10 th, 20 th, 30 th, 40 th, 50 th,……10 th percentiles

37. Examining Distributions: Quantitative …  Numbers Measures of central tendency; mean, median, mode Measures of variation around mean – variance, standard deviation, standard error of mean Measures of variation around median – percentiles, quintiles, quartiles  Graphs Histograms The five-number summary  Boxplot

38. Histogram Outliers??

39. Histograms …  Used for interval and ratio data.  A histogram is a graph in which each bar (horizontal axis) represent a range of numbers called interval width. The vertical axis represents the frequency of each interval.  There are no spaces between bars.  Histogram is useful for graphic illustration of one group.

40. Box plot: 5 – number summary Range = Max - Min IQR = Q3 – Q1 Whiskers Q3 Q1 Median/Q2 Inner fence Outliers 1 st 100 th

41. Box plot of change in pain score

42. Box Plots …  Used for interval and ratio data.  Uses the five-number summary measures Median, Q1, Q3, minimum and maximum.  It is useful in detecting outliers  It is useful to illustrate the distribution of more than on group.

43. What are outliers … ? Outliers are extreme data values that fall outside of distribution of the data set.

44. Box plot: 5 – number summary IQR = Q3 – Q1 Whiskers Q3 Q1 Median/Q2 Inner fence 1 st 100 th

45. 1.5  IQR Criterion for Outliers  Interquartile range (IQR) is the distance between the first and third quartiles. IQR = Q 3 – Q 1  From data Q 1 = 59 yrs, Q 3 = 70 yrs, IQR = 70 – 59 = 11 1.5  IQR = 1.5  11 = 16.5 Q 1 – IQR = 59 – 16.5 = 42.5 Q 3 + IQR = 70 + 16.5 = 86.5  From data: Min= 44 and Max = 82

46. Properties of quartiles, quintiles…  It is used for interval or ratio data.  It is resistant (insensitive) to extreme values.  It is useful for summarising skewed data.

47. How to deal with skewed data  Transform the data: Square/square root – (Poisson) count data Log(x) or ln(x) – data is skewed toward right Reciprocal (1/X) - data is skewed toward left  Transformation: Make skewed data more symmetric Makes distribution more normal Stabilize variability Liberalize a relationship between two or more variables  Show summary stat in original but analyse on the transformed data

48. Summary of what we have learned ….  Always plot your data: make a graph, e.i. histogram, box plot  Look for overall pattern (shape, centre and spread) and for striking deviations such as outliers  Check to see if overall pattern of distribution can be described by normal distribution.  If not uniform, transform data to make skewed data more symmetric  Calculate an appropriate numerical summary to describe centre and spread