24 Donovan pages 116 EE
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1. SURVIVAL ANALYSIS Objectives e Simulate the fates of 25 individuals over a 10 day period e Calculate the Kaplan Meier product limit estimate e Graphically analyze the Kaplan Meier survival curve e Assess how sample size affects the Kaplan Meier estimate e Assess how censorship affects the Kaplan Meier estimate Suggested Preliminary Exercise Life Tables and Survivorship Curves INTRODUCTION A population of black bears has been surveyed for 10 years and ecologists note that the number of bears in the population has declined over this time frame Why Changes in numbers of individuals over time can be directly traced back to the population s birth death immigration and emigration rates The popula tion may have declined because the birth rate dropped the death rate increased immigration dropped or emigration increased A combination of any or all of these factors may also be responsible for the decline Mortality and its counter part survival are keys to the demographic equation for all organisms How do ecologists estimate these two important parameters In this exercise we ll explore one method for estimating survival In your life table exercise you tracked the fates of individuals over time not ing how many individuals in the cohort were still alive at each time step and then calculated the survivorship schedule and survival probabilities from your data Suppose we followed a cohort of 100 newborns over time2. if you know the number of deaths at a specific time and the number of individuals at risk at that same time The number of deaths divided by the number at risk gives the conditional probability of mortality so 1 minus that value is the conditional probabil ity of survival In cell B39 we used the formula PRODUCT B 38 B38 The unconditional probability of survival is the probability of surviving to a particu lar time It is calculated in Equation 2 as the cumulative product of the conditional prob abilities i dj P Tl 3 j 1 J In cell B40 enter the formula B 4 B9 The symbol means raises the value in cell C4 the survival probability to the num ber of days under consideration In cell B41 enter the formula B39 1 B9 to obtain the daily survival estimate for day i Remember that the P gives the probability of surviving to a specific time period To convert the P to daily survival probabilities take the appropriate root For example take the third root of P for day 3 the seventh root of P for day 7 and so on to obtain the daily survival estimate To obtain roots in spreadsheets use the exponent form with the exponent as a fraction In cell C35 enter the formula B35 B36 B37 Remember that the number of individuals at risk are those currently alive and not cen sored Your spreadsheet should now look something like Figure 7 but with the exception of Row 40 your numbers will likely be different rr a ae ee
3. The Kaplan Meier method 1958 involves tracking the fates of individuals over time and estimating how long it takes for death to occur The method has been applied broadly to measure how long it takes for any specific event to occur such as the time it takes until death the time until a cancer patient recovers from a treatment the time until an infection appears the time until pollination occurs and so on The Kaplan Meier method is conceptually similar to life table calculations because you keep track of the number of individuals alive and the number of deaths that occur over intervals of time Specifically you count the number of individuals who die at a certain time and divide that number by the number of individuals that are at risk alive and part of the study at that time If we do this for each time period in the study we will be able to compute two survival probabilities the conditional survival probability and the uncon ditional survival probability We will describe how each is computed with a brief example Suppose you initiate a study on beetle mortality and track 20 individuals over 5 days each day recording the number of deaths and the number of individuals still alive Let s also suppose that some of your population decides to emigrate out of the population so you can no longer track their fates The data you collect are ean De Emigrants Deaths Now let t be a particular time period such
4. carefully noting when deaths occurred We start with S 100 count individuals again at the next time step S and then at time step S Suppose S 40 and S 10 The survivorship schedule see Exercise 12 Life Tables Survivorship Curves and Population Growth tells us that the probability that an individual will survive from birth to time x Thus the probability of surviving to age 1 is S S 40 100 0 4 and the probability of surviving from birth to age 2 is S S 10 100 0 1 Age specific survival probabilities in contrast tell us the probability that an individual will survive from one age to the next such as the probability that an individual alive in time S will be alive at time S In life table calculations the age specific survival probability is calculated as g 1 l In our example the probability that an indi vidual of age S will survive to age S is 0 10 0 40 0 25 The life table cohort 312 Exercise 24 analysis is one way of calculating survival However this method is not always possi ble to use especially if the organisms of interest are long lived Fortunately alternatives for estimating survival exist Kaplan Meier Survival Analysis When the research question can be posed as how long does it take until death occurs the Kaplan Meier survival analysis also known as the Kaplan Meier product limit estimate or the Kaplan Meier survival curve can be used to estimate survival
5. P 1 3 20 3 20 0 15 0 15 0 85 0 85 20 3 1 16 4 16 0 25 0 25 0 75 0 85 0 75 6375 2 16 4 0 12 2 12 0 16 0 16 0 84 0 85 0 75 0 84 54 1 12 2 1 9 1 9 0 11 0 11 0 89 0 85 0 75 0 84 0 89 48 0 1 0 0 2 9 1 8 2 8 0 25 0 25 0 75 0 85 0 75 0 84 0 89 0 75 36 Figure 2 Notice that P decreases with each day because the probability of living to a given period must decrease as ever greater time periods are considered Sometimes ecolo gists are interested in expressing P as a daily probability To obtain a daily survival estimate you would take the appropriate root For example P 0 36 on day 5 in Figure 2 This gives the probability that an individual will survive through day 5 What would daily survival be to obtain P 0 36 on day 5 A daily probability of x would have to yield 0 36 when multiplied by itself once for each day so x 0 36 By taking the fifth root of 0 36 you could solve for x The spreadsheet formula is 0 36 1 5 Kaplan Meier Survival Curves The results of the Kaplan Meier analysis are often graphed graphs are known as the Kaplan Meier survival curves Figure 3 Comparing the survival curves of two dif ferent populations can yield insightful information about the timing of deaths in 314 Exercise 24 INSTRUCTIONS A Set up the model pop ulation 1 Op
6. aily survival Figure 6 B a a Coe 6 a a Enter 25 in cell B35 The number at risk on day 1 is 25 because we started with a sample size of 25 In cell B36 enter the formula COUNTIF B10 B34 D The number of deaths on day 1 is the number of D s that appear for the 25 individuals 4 In cell B37 enter a for mula to count the number of censored observations on day 1 5 In cell B38 enter a for mula to compute the con ditional probability of sur vival P 6 In cell B39 enter a for mula to compute the unconditional probability of survival P 7 In cell B40 enter a for mula to compute the expected P for day 1 given the survival param eter in cell B4 8 In cell B41 enter a for mula to compute the actual daily survival for each P 9 In cell C35 compute the number of individuals at risk for day 2 10 Select your formulae from steps 3 8 and copy them across to column K 11 Save your work Survival Analysis 317 In cell B37 enter the formula COUNTIF B10 B34 C The number of censored observations on day 1 is the number of C s that appear for the 25 individuals In cell B38 enter the formula 1 B36 B35 This is the spreadsheet version of Equation 1 1_4i P 1 The conditional probability of survival is the probability of survival to a particular time period given that you survived to the previous time This probability is easy to calculate
7. as 1 day d be the number of deaths at time t n be the number of individuals at risk at the beginning of time t The conditional survival probability P is the probability of surviving to a specific time given that you survived to the previous time this is similar to the age specific survival probabilities in the life table P is computed as P 1 4 Equation 1 The term d n gives the number of individuals that die in time step i divided by the number of individuals still alive and still in the population the number at risk This is the conditional mortality probability or the probability that an individual will die during that time step Since survival can be computed as 1 minus mortality Equation 1 gives the conditional survival probability Because we started with a population of 20 individuals the number at risk for death at the beginning of day 1 is 20 During that day 3 individuals died so the conditional mortality probability is 3 20 0 15 and the conditional survival probability is 1 0 15 0 85 Now let s consider day 2 At the beginning of day 2 there are only 16 individu als at risk Three individuals died the previous time step and one left the population through emigration The individual that left the study is called a censored observation 313 Survival Analysis Individuals that die in the previous time step as well as censored individuals cannot be considered at risk so on day 2 only 16 individuals are
8. at risk On day 2 4 deaths occurred so the conditional mortality probability is 4 16 0 25 and the conditional sur vival probability is 1 0 25 0 75 The rest of the computations are shown in Figure 1 Deaths at risk 3 20 0 15 4 16 0 25 1 0 15 0 85 1 0 25 0 75 1 0 16 0 84 1 0 11 0 89 1 0 25 0 75 2 12 0 16 1 9 0 11 2 8 0 25 Figure 1 The unconditional survival probability P is the probability of survival from the start of the study to a specific time this is similar to the survivorship schedule in the life table The unconditional probability is equal to the cumulative product of the con ditional probabilities which is why the Kaplan Meier method is sometimes called the Kaplan Meier product limit estimate The equation can be expressed as i dj Py J114 j 1 J where the IT symbol means multiply all of the individual conditional probabilities together The computations are shown in Figure 2 For day 1 the unconditional survival probability is the same as the conditional sur vival probability P for day 2 gives the probability that an individual at the start of the study will survive through day 2 This is obtained by multiplying the conditional sur vival probability for day 1 by day 2 since both conditions must be met in order for an individual to be alive at the end of day 2 Equation 2 B C D E F G Emigrants Deaths at risk Deaths at risk Pe
9. ate in the study depending on the time of year your study is being conducted Compare how early censorship and late censorship affect P and P Set cell B6 0 5 to assess early censorship the remaining cells should be 0 Then set cell K6 0 5 the remaining censorship probabilities should be 0 Describe your results in terms of P and its standard deviation In the spreadsheet model we simulated the fate of individual s death or sur vival by linking a random number to a daily survival probability in cell B4 Thus we assumed that for each day an individual had the same probability of surviving as any other day What happens to the Kaplan Meier estimates when survival probabilities vary over the course of the study Modify your model to include this change and discuss your results in graphical form For example establish different daily survival probabilities in cells B4 K4 and adjust the for mulae in cells B10 K34 so that the daily survival probability reflects your new entries in cells B4 K4 Advanced How does sample size affect both P and P Modify your model and compare results when the sample size is increased from 25 to 50 individuals LITERATURE CITED Kaplan E L and P Meier 1958 Nonparametric estimation from incomplete obser vations Journal of the American Statistics Association 53 457 481
10. e ee ee eee Se a e E E E E A E E E E 36 deatns 3 2f of 4 of of of 1f ol 3z censoed 2 a 3f af of of of ol 38 Conditional P osef oof tf oo17 0 9 af af 1f osf 1 0 523 0 349 al 0523 40 Expected survival 09 0 87 0720 0 656 0590531 0478 0 43 0 67 0 349 Figure 7 318 Exercise 24 D Create graphs 1 Graph P P and expected P as a function of time Interpret your graph 2 Press F9 to generate a new simulation How do your results appear to change with each new simulation E Track 100 simula tions 1 Set up new headings as shown in Figure 9 but extend the trials to 100 cell M109 and the days to 10 cell W9 2 Record a macro to track P for 100 trials logging your results in cells N10 W109 Use the line graph option and label your axes fully 1 a Ne A 0 7 ODA x 0 6 Conditional Pc o Unconditional Pu 0 4 A a Expected Survival Probability of survival Oo ui Figure 8 Your graph will look different than the Kaplan Meier survival curve because the points are connected differently However the graphs are interpreted the same way Note that the expected P is a straight line because we set the daily survival probability as a constant over time Sharp drops in the P line indicate more mortality on a given day and shallow drops in a line indicate fewer deaths occurrin
11. en a new spreadsheet and set up column head ings as shown in Figure 4 Kaplan Meier Survival Kaplan Meier Survival Curve Curve 1 1 08 0 8 0 6 0 6 3 3 o4 a o4 0 2 4 024 0 T T 1 0 T T 1 0 2 4 6 0 2 4 6 Day Day Figure 3 Kaplan Meier survival curves for a hypothetical population The unit time is plotted on the x axis P is plotted on the y axis In Kaplan Meier curves the raw data are plotted as in the graph on the left then the data points are connected with horizontal and vertical bars as shown on the right Large vertical steps downward indicate a large num ber of deaths in the given time period while large horizontal steps indi cate few deaths have occurred during an interval response to different environmental conditions Often in the literature you will see the survival curves for two different populations on the same graph so that you can com pare the two easily PROCEDURES The method outlined by Kaplan and Meier 1958 is one of the most referenced papers in the field of science suggesting that is has played an important role in ecology and other sciences since its publication The goal of this exercise is to set up a spreadsheet model of the Kaplan Meier product limit estimate and to learn how censored obser vations and sample size affect the survival probabilities As always save your work frequently to disk ANNOTATION We ll track 25 individuals f
12. etches of slightly sloping or horizontal lines indicate What do steeply sloping vertical drops indicate What level of daily survival is needed to ensure that the population will persist for 10 days Set up your spreadsheet as shown Enter the expected P s for each level of daily survival given in cells A45 A53 For example cell B45 should compute P for day 1 when the daily survival is 0 1 Under what conditions is a population likely to persist for at least 10 days Graph your results __ aAa ee a eee eee n i sf k 44 Daily Survival Expected P 45 sl of S T oOo ajo ooo e e e ee ee ee ee ee eee a RSE NN SPS E a ost 50 guy i The Kaplan Meier estimate is often used because uncooperative individuals can be taken out of the picture For example individuals that fly away from your study plot are censored observations and can be subtracted from your at risk population Compare your model results to a population where censored observations are absent cells B6 K6 0 Erase your macro results cells N10 W109 then run your macro again under the new conditions Compare the average P and the standard deviations of the trials Under some conditions censored observations may occur early in the study and under some conditions censored observations may occur late in the study For example dispersal of individuals out of your study population may occur early or l
13. g during a particular inter val Figure 8 shows few no deaths actually occurred from Day 5 to Day 8 Your results should vary from simulation to simulation This is due to the random num ber function changing the data set and it is also due to the fact that our population con sists of only 25 individuals so there is some demographic stochasticity in this model In order to fully understand how P and P behave over the 10 day period we need to run several simulations and track our results We will do that in the next step Figure 9 Open up the macro function as described in Exercise 2 or your user s manual Once you have assigned a shortcut and the macro is in Record mode perform the follow ing steps e Select cells B39 K39 Copy e Select cell N9 Open Edit Find e Leave the Find What box empty and search by columns Select Find Next then Close Your cursor should move down to cell N10 3 Use the AVERAGE func tion in cells N110 W110 and STDEV function in cells N111 W111 to compute the average P and standard deviation for the 100 trials 4 Graph the average P for each day 5 Add error bars to your graph First divide each standard deviation by 2 in cells N112 W112 6 Save your work Survival Analysis 319 e Open Edit Paste Special and select the Paste Values option Click OK e Select Tools Macro Stop Recording Run your macro until 100 trials have been co
14. ility to 0 1 for all days Later in the exercise you will change these values to determine how censored observations and the time at which they occur affect survival probability estimates In cell B10 enter the formula IF RAND lt B 6 C IF RAND gt B 4 D 1 Copy your formula down to row 34 The formula in B10 will assign a fate to individual 1 on day 1 The individual will be either alive 1 censored C or dead D The formula contains two IF functions and a RAND function so it is a nested formula Remember that the IF function consists of three parts separated by commas In the first part of the function you specify a criteria If the crite ria is true the spreadsheet will do or carry out whatever you specify in the second por tion of the function If the criterion is false the spreadsheet will carry out what you spec ify in the third portion of the function Let s review the B10 formula carefully The criterion is that a random number the RAND portion of the formula is less than the value in cell B6 the probability of being censored on day 1 If the criterion is true the individual is censored and the spreadsheet will return the letter C If the criterion is false the individual is not censored and the second IF function will be computed The second IF function tells the spreadsheet to evaluate whether a random number between 0 and 1 is greater than the value in cell B4 the true but unknown to you the
15. mputed Your formula for day 1 should be AVERAGE N10 N109 This gives the average unconditional probability that an individual will survive past day 1 The standard devi ation is computed as STDEV N10 N109 You will want to divide this number by 2 for graphing purposes in the next step Use the column graph option Your graph should resemble Figure 10 without the error bars Unconditional P Survival to Day X over 100 Trials 1 0 8 0 6 Average 0 4 0 2 0 T T T T T T T ji Figure 10 To add error bars click on the columns in the graph to select them Then go to Format Selected Data Series Y Error Bars Select the Custom option Click on the Display Both option Place your cursor in the box labeled then use your mouse to select the standard deviations for your 100 trials divided by 2 cells N112 W112 Do the same for the box labeled Click OK and your graph should be updated Patamu Asa YE Bers Data Lakak Sarima Crdier options Tinpiay ae Bath Phun Pirun Hora Encor aart F fhad vaba z a E Barcuntaga ko ak pedo 4 seeded gor Foam f e Figure 11 320 Exercise 24 QUESTIONS 1 N ies o1 6 Interpret the Kaplan Meier conditional and unconditional probabilities graph e g Figure 8 What do long str
16. or 10 days and keep track of their fates over time Row 10 will track Individual 1 s fate Row 11 will track Individual 2 s fate and so on to Row 34 a ee ee ee eee eee Survival Analysis Total sample 6 Prob of censor 7 8 9 Individual Figure 4 2 Set up a linear series from 1 to 25 in cells A10 A34 3 In cell B4 enter a value for the probability that an individual will survive each 24 hour period daily survival 4 Enter the number of individuals in the initial population in cell B5 5 In cells B6 K6 enter a value for the probability that an individual in the population will be cen sored on a given day 5 Save your work B Simulate fates of individuals over time 1 In cells B10 B34 enter a formula to assign a fate to each individual for day 1 2 In cell C10 enter a for mula to assign a fate to individual 1 for day 2 Survival Analysis 315 In cell A10 enter the value 1 In cell A11 enter the formula 1 A10 Copy this formula down to cell A34 Enter the value 0 9 in cell B4 In reality you wouldn t know what this number is you are using the Kaplan Meier method to estimate this parameter Enter the value 25 in cell B5 Enter the value 0 1 in cells B6 K6 This is the probability that an individual will leave the study on any given day so that its fate cannot be tracked over time For now we set that probab
17. researcher daily survival probability If the random number is greater than the sur vival probability the individual will die the spreadsheet will return the letter D If the random number is less than the value in cell B4 the spreadsheet will return the num ber 1 indicating that the individual survived that day When you copy your formula down for the 25 individuals in the population you should see that some individuals die and some become censored Press F9 the calculate key to generate new fates for individuals in the population In cell C10 enter the formula IF OR B10 D B10 C B10 IF RAND lt C 6 C IF RAND gt B 4 D 1 Don t be intimidated by the length of this formula If the individual in cell C10 died or was censored on day 1 we want to return a blank cell i e two double quotes If the individual survived day 1 then we want to know what happened on day 2 The formula in cell C10 is another nested IF function There 316 Exercise 24 3 Select cell C10 and copy its formula across to cell K10 Modify the formula in each cell to reflect the probability of censorship for the appropriate day 4 Select cells C10 K10 and copy the formula down to row 34 5 Save your work C Compute survival probabilities 1 Set up new headings as shown in Figure 6 2 In cell B35 enter the number of at risk individ uals in the population on day 1 3 In cell B36 enter a for m
18. ula to count the number of deaths on day 1 are multiple criteria however in the first IF function and these criteria are given with an OR function The OR function is used to evaluate whether the value in cell B10 is D or C or If any one of those three conditions is true the spreadsheet will return a blank or If none of the conditions is true the individual must have survived day 1 and the second IF function is computed it has the same form as the formula in cell B10 with the spreadsheet again returning a value of C D or the number 1 Double check your formulae They should read as follows e In cell D10 IF OR C10 D C10 C C10 IF RAND lt D 6 C IF RAND gt B 4 D 1 e In cell E10 IF OR D10 D D10 C D10 IF RAND lt E 6 C IF RAND gt B 4 D 1 e In cell F10 IF OR E10 D E10 C E10 IF RAN D lt F 6 C IF RAND gt B 4 D 1 and so on Your spreadsheet should now resemble Figure 5 although the fates of your individuals will likely be different than that shown Figure 5 The first calculations in the Kaplan Meier estimate involve counting the number of individuals at risk still alive during each day and to count the number of deaths that occur each day E FG H J K Unconditional P 40 Expected survival D
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