UNIVERSITY OF MASSACHUSETTS Department of Biostatistics and Epidemiology BioEpi 540W - Introduction to Biostatistics Fall 2004

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1 UNIVERSITY OF MASSACHUSETTS Department of Biostatistics and Epidemiology BioEpi 50W - Introduction to Biostatistics Fall 00 Exercises with Solutions Topic Summarizing Data Due: Monday September 7, 00 READINGS. Text (Rosner B. Fundamentals of Biostatistics, 5 th Edition) Chapters and.. Study Guide (Rosner B. Study Guide for Fundamentals of Biostatistics, 5 th Edition) Chapter. EXERCISES:. For each of the following variables indicate whether it is quantitative or qualitative and specify the measurement scale that is employed when taking measurements on each: (source: Daniel, page, problem #6.) a) Class standing of members of this class relative to each other b) Admitting diagnosis of patients admitted to a mental health clinic c) Weights of babies born in a hospital during a year d) Gender of babies born in a hospital during a year e) Range of motion of elbow joint of students enrolled in a university health sciences curriculum f) Under-arm temperature of day-old infants born in a hospital

2 . Using the data below (source: Daniel, 6 th edition page 30, problem.3.5), a. Construct a stem and leaf display. b. Construct a frequency table with columns for frequency, relative frequency, cumulative frequency, and cumulative relative frequency. c. Construct a histogram. d. Construct a frequency polygon 3. Data were recorded on the age in years and height in cm of 0 high school students in a classroom. Females Males Age Height Age Height

3 3 a. Create a frequency table for age, with columns for frequency, relative frequency, cumulative frequency, and cumulative relative frequency. b. Create a histogram for age. c. For each sex, create a stem-and-leaf display for height. What does a comparison of the displays suggest about the students? d. For each sex, create histograms for height using the same scale.. Let x =3, x =, x 3 =, and x =6 a. Express the following sum in sigma notation and evaluate numerically. (x + x + x 3 + x ) b. Express the following sum in sigma notation and evaluate numerically. x + x + x 3 + x c. Evaluate the following numerically. Σ (X i ) for i=. d. Evaluate the following numerically. Σ 3X i for i=. 5. a. The following are behavioral ratings as measured by the Zang Anxiety Scale (ZAS) for 6 persons with a diagnosis of panic disorder: Compute the mean, median, mode, range, variance, and standard deviation, and the 5th and 75th percentiles.

4 b. The following are behavioral ratings as measured by the Zang Anxiety Scale (ZAS) for healthy controls: Compute the mean, median, mode, range, variance, and standard deviation, and the 5th and 75th percentiles. c. Construct Box and Whisker plots using the data from parts "a" and "b". In one or two sentences, compare the two groups of sentences. 6. The following table shows the age distribution of cases of a certain disease reported during a year in a particular state. Age Number of Cases TOTAL 75 6a. Construct a frequency table with columns for class endpoints, class midpoint, frequency, relative frequency, cumulative frequency, and cumulative relative frequency. 6b. Construct a cumulative relative frequency plot of the data. Use this plot to estimate the 0th, 5th, 50th, and 75th percentiles. 6c. Compute the mean, median, variance, and standard deviation.

5 5 7. For women undergoing in vitro fertilization, various therapies are used to stimulate the ovaries. In one study comparing the effectiveness of a new hormone therapy on three groups of women with different types of fertility problems, an outcome of interest is the number of oocytes that 'ripened'. Some summary statistics on the number of ripened oocytes per woman for each of the three groups are reported below. Group Statistic 3 n 38 9 mean median P5 5 5 P75 minimum 5 maximum 0 3 7a. Compute box and whisker plots for the three groups. 7b. In your opinion, which statistics are best for comparing these three groups? Why?

6 6 SOLUTIONS #. a. Qualitative - ordinal b. Qualitative - nominal c. Quantitative - ratio d. Qualitative - nominal e. Quantitative - ratio f. Quantitative - interval #. How to Create a STATISTIX Data Set Containing This Information STEP. Enter the data. ) Begin the software program STATISTIX ) Click x on DATA. 3) Click x on INSERT ) Click x on VARIABLES. 5) Click x inside the NEW VARIABLE NAMES dialog box. Type X <enter>. 6) Click x on OK At this point you will see a spread sheet with space for only one record. You ll need to instruct STATISTIX that you actually have a data set of 5 records 7) Click x on DATA 8) Click x on INSERT 9) Click x on CASES 0) Click x inside the FIRST NEW CASE NUMBER dialog box Type <enter> ) Click x inside the NUMBER OF CASES TO INSERT dialog box. Type <enter> 7) To enter your data by rows: Type in the first value. Press the RIGHT arrow. Type in the second value. Press <enter>. All other values are entered the same as the second value. To enter your data by columns: Type in the first value. Press the DOWN arrow. Type in the second value. Press <enter>.

7 7 All other values are entered the same as the second value. Continue until all values of one column are entered. Use the arrow keys to get to the first record in the next column and proceed like with the first column. You should have 5 records with values per record. If you have 6 records, click x on the record number 6. Click x on EDIT. Click x on CUT. STEP. Save the data. ) Click x on FILE. ) Click x on SAVE AS. 3) *.sx should be highlighted in the FILE NAME dialog box. Type unit3ex.sx <enter>. #a. Here is the stem and leaf diagram I constructed Stem Leaf Other groupings for the stem are okay.

8 8 How to Request a Stem and Leaf Plot in STATISTIX Following assumes that you are already in STATISTIX and have already opened unit3ex.sx ) Click x on STATISTICS ) Click x on SUMMARY STATISTICS. 3) Click x on STEM AND LEAF PLOT ) Click x on the variable X Then click on the RIGHT ARROW 5) Click x on OK You should get the following STEM AND LEAF PLOT OF X LEAF DIGIT UNIT = MINIMUM REPRESENTS. MEDIAN MAXIMUM STEM LEAVES (7) CASES INCLUDED 0 MISSING CASES To the reader: Can you guess what the numbers at the far left are telling you? Hint Read from top to bottom, then from bottom to top!

9 9 How To Insert Text STATISTIX Results into a WORD Document Following assumes that you are in STATISTIX and have just gotten the plot ) Click x on FILE ) Click x on SAVE AS. 3) (Check to be sure results are being saved to the directory of your choosing) ) Type unit3exa.txt in the file name dialog box 5) (Minimize the STATISTIX window so that you can work in WORD) Next assumes that you are in WORD 6) Position cursor to where you want the results to be placed 7) Click x on INSERT 8) Click x on FILE 9) Using the LOOK IN feature, position yourself in the directory containing your results 0) Click x on unit3exa.txt

10 0 #b. Here is what I constructed by hand Class Relative Cumulative Cumulative Interval Frequency Frequency Frequency Rel. Frequency TOTAL Other class intervals are okay.

11 How to Request a Frequency Distribution in STATISTIX Following assumes that you are already in STATISTIX and have already opened unit3ex.sx AND that you have just finished your stem and leaf diagram ) Click x on WINDOW ) Click x on unit3ex.sx. At this point, STATISTIX should have returned you to your spreadsheet of data 3) Click x on STATISTICS 3) Click x on SUMMARY STATISTICS ) Click x on FREQUENCY DISTRIBUTION ) Click x on the variable X Then click on the RIGHT ARROW 5) Click x on OK You should get the following FREQUENCY DISTRIBUTION OF X CUMULATIVE VALUE FREQ PERCENT FREQ PERCENT TOTAL

12 #c. How to Request a Histogram in STATISTIX Following assumes that you are already in STATISTIX and have already opened unit3ex.sx AND that you have just finished some other description or analysis ) Click x on WINDOW ) Click x on unit3ex.sx. 3) Click x on STATISTICS ) Click x on SUMMARY STATISTICS 5) Click x on HISTOGRAM 6) Click x on the variable X Then click on the RIGHT ARROW 6) Click x on OK You should get the following (note on titles below ) 0 Histogram for #c 8 Frequency X To get the titles as shown here 7) Click x on RESULTS 8) In the TOP TITLE dialog box, type Histogram for #c

13 3 How To Insert GRAPHICAL STATISTIX Results into a WORD Document Following assumes that you are in STATISTIX and have just gotten the plot ) Click x on FILE ) Click x on SAVE AS. ) (Check to be sure results are being saved to the directory of your choosing) ) Type unit3exc.emf in the file name dialog box 5) (Minimize the STATISTIX window so that you can work in WORD) Next assumes that you are in WORD 6) Position cursor to where you want the results to be placed 7) Click x on INSERT 8) Click x on PICTURE 9) Click x on FROM FILE 0) Using the LOOK IN feature, position yourself in the directory containing your results ) Click x on unit3exc.emf

14 #d. How to Request a Cumulative Distribution in STATISTIX Following assumes that you are already in STATISTIX and have already opened unit3ex.sx AND that you have just finished some other description or analysis ) Click x on WINDOW ) Click x on unit3ex.sx. 3) Click x on STATISTICS ) Click x on SUMMARY STATISTICS 5) Click x on HISTOGRAM 6) Click x on the variable X Then click on the RIGHT ARROW 7) CLICK x on the CUMULATIVE DISTRIBUTION option 8) Click x on OK You should get the following (except that I changed the title a bit ) 00 Cumulative Distribution 80 Percent X

15 5 #3a. Relative Cumulative Cumulative Age Frequency Frequency Frequency Rel. Frequency TOTAL 0.00 #3b. Histogram of Age Frequency Age, years

16 6 #3c. Here is what I get by hand (I like it better than the STATISTIX output below) Females Stem Males How To Supply Labels for Values of Discrete Variables (so that output is easier to read!) Following assumes that you are in STATISTIX and you wish to assign labels to the values 0 and for the variable SEX so that 0=FEMALE and =MALE ) Click x on WINDOW ) Click x on unit3ex3 spreadsheet. 3) Click x on DATA ) Click x on LABELS 5) Click x on VALUE LABELS 6) Click x on the variable SEX and RIGHT ARROW it to the SOURCE VARIABLE box 7) In the VALUE box type 0. Below this, in the LABEL box type 0=Female 8) In the VALUE box type. Below this, in the LABEL box, type =Male. 9) Click x on SAVE. 0) Click x on CLOSE.

17 7 This is what I get in STATISTIX STEM AND LEAF PLOT OF HEIGHT FOR SEX = 0=female LEAF DIGIT UNIT = MINIMUM REPRESENTS 5. MEDIAN MAXIMUM 7.00 STEM LEAVES () CASES INCLUDED 0 MISSING CASES STEM AND LEAF PLOT OF HEIGHT FOR SEX = =male LEAF DIGIT UNIT = MINIMUM REPRESENTS 67. MEDIAN MAXIMUM STEM LEAVES () CASES INCLUDED 0 MISSING CASES Males tend to be taller than females #3d. Class FEMALES MALES Interval Freq. Re. Freq. Freq. Rel. Freq

18 8 How To Select A Subset of the Data for Analysis We wish to construct separate histograms, first for females, then for males. STATISTIX is a bit awkward in this. To Select Females, STATISTIX requires that you omit Males ) Click x on WINDOW ) Click x on unit3ex3 spreadsheet. 3) Click x on DATA ) Click x on OMIT/SELECT/RESTORE CASES 5) Type Omit sex= 6) Click x on GO 7) Click x on CLOSE Construct your histogram using instructions per above. Take care to title it clearly. Before you can select males, you must restore the entire data set ) Click x on WINDOW ) Click x on unit3ex3 spreadsheet 3) Click x on OMIT/SELECT/RESTORE CASES ) Type restore To Select Males, STATISTIX requires that you omit Females ) Type Omit sex=0 ) Click x on GO 3) Click x on CLOSE

19 9 This is what I get in STATISTIX Histogram of Heights - FEMALES 5 Frequency HEIGHT Histogram of Heights - MALES 5 Frequency HEIGHT

20 0 A. ( ) X + X + X 3 + X = Σ X i = = = = 96. i ( ) B. X + X + X + X = Σ X 3 = 3 i= i = = 6. C. Note: Σ( ) = ( 3 ) + ( ) + ( ) + ( 6 ) X i i= Σ = = = 38. ( Xi ) = Σ [ Xi Xi + ] i= i= = Σ X - Σ X + Σ i i i= i= i= = 6 - = 38. ( )( ) + ( )( )

21 D. Σ 3 Xi = 3 Σ X i i= i= ( ) = 3 = 5 A stem and leaf diagram might come in handy: MEAN MEDIAN 6 x = Σ X n i= i ( 56 ) =. 6 so = 5. = n x n First solve = = 35. th th Median is midpoint of 3 and observation. ~ x = ( + 3) so ~ x = MODE This sample is tri - modal 38, 5, RANGE Maximum - Minimum = 69 3 so range = 38

22 VARIANCE Let s save ourselves the trouble of a very long brute force formula by using the formula for grouped data. Let j index the unique values. There are unique values. j X j f j ( x j x) ( j ) f x x j TOTALS S j= = ( j ) Σ f x x j Σ j= f j = So S = Standard deviation S = S So S = 89.

23 3 5th Percentile First solve (.5) (n) = (.5) (6) = 6.5 So 5th percentile is the 7 th observation P 5 = 38 75th Percentile First solve (.75) (n) = (.75) (6) = 9.5 So 75th percentile is the 0 th observation P 75 = 5 5B MEAN x = Σ X ( ) i = 568 = 7. 0 So x = 7. 0 n i= MEDIAN n + + Solving = = Median is the th observation. So ~ x = 6 MODE mode = 5 RANGE Maximum - Minimum = 3 5. So Range = 9

24 Variance There are 6 unique values. j X j f j ( x j x) ( j ) f x x j TOTALS 9 98 S j= = ( j ) 6 Σ f x x j 6 Σ j= Standard deviation f j 67 = So S = S= S = 835. So S= 89. 5th Percentile Solving (.5) (n) = (.5) () = 5.5 So 5th percentile is 6th observation P 5 = 5 75th Percentile Solving (.75) (n) = (.75) () = 5.75 So 75th percentile is 6th observation P 75

25 5 5C. REMINDER - Use the same scale when comparing two groups. Group Patients Controls Mean Median 6 P P Interquartile Range (IQR) 3 3 P5-(.5)(IQR) P75+(.5)(IQR) * Min 3* 5* Max 69* 3 *=Whisker Notes on Whiskers ) IF P 5 - (.5) (IQR) < minimum, use minimum instead ) IF P 75 + (.5) (IQR) > maximum, use maximum instead Patients with panic disorder have Exercise ZAS scores #C that - Box are and higher Whisker than those Plotof controls. As well, ZAS scores of patients with panic disorder have more variability ZAS Score HEALTHY PANIC Healthy (n=), Panic Disorder (n=6)

26 6 6A. Class Class Relative Cumulative Cumulative Endpoints Midpoint Frequency Frequency Frequency Relative Freq TOTALS.000 6B. 00 Exercise #3b - Cumulative Rel. Frequency 80 Percent AGE Estimates are P 0 = 7 P 50 = 36 P 5 = 6 P 75 =.5

27 7 6C. Midpoint X j Frequency f j X j f j ( x x) ( ) j f j x j x MEAN x 6 Σ fx j j j= 680 = 6 = So x = Σ f 75 j= j 357. MEDIAN Note to reader I ve consulted a number of texts on this. There is no single correct answer. With interval data, whatever median you calculate is an approximation. Here is what is suggested in Think and Explain with Statistics (Lincoln E. Moses, page 6) First solve n = = 38 th observation Examination of the table reveals that the 38 th observation is in the interval 35 to.99 Set the following quantities: The letter l = lower limit of interval = 35 The letter u = upper limit of interval =.99 R = cumulative frequency up to the lower limit of interval = 35 M = # observations contained in interval = N = total # observations = 75 An approximate solution for the median is calculated as L NM ~ N/ - R x= l + ( u l) = 35 + M QP O L NM 75/ - 35 O QP ( ) = or 37

28 8 VARIANCE S j= = ( j ) 6 Σ f x x 6 j Σ j= f j = so S = Standard deviation S = S so S = 30. 7A. Remember to use the same scale. 3 median mean left box edge = P right box edge = P 75 IQR = P 75 -P P 5 -(.5)(IQR) P 75 +(.5)(IQR) left whisker 5 right whisker 0 3

29 9 7B. When data are skewed by extreme values, medians and quartiles give a better feel for the bulk of the data than do means and standard deviations. This example also illustrates that, as sample size increases, the range can only increase. Notice that the extreme value of 0 occurred in the sample with the largest sample size.