17 Probit Analysis

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MEET MTB UGUIDE 1 SC QREFUGUIDE 2INDEXCONTENTS HOW TO USE

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RSPROBIT.MK5 Page 1 Friday, December 17, 1999 12:39 PMUsers Guide 2 17-1

MEET MTB UGUIDE 1 SC QREFUGUIDE 2INDEXONTENTS HOW TO USE

Probit Analysisn Probit Analysis Overview, 17-2

n Probit Analysis, 17-2

Chapter 17 Probit Analysis Overview

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Probit Analysis OverviewA probit study consists of imposing a stress (or stimulus) on a number of units, then recording whether the unit failed or not. Probit analysis differs from accelerated life

Pro

RSPROBIT.MK5 Page 2 Friday, December 17, 1999 12:39 PMMINITAB Users Guide 2


testing (page 16-6) in that the response data is binary (success or failure), rather than an actual failure time.

In the engineering sciences, a common experiment would be destructive inspecting. Suppose you are testing how well submarine hull materials hold up when exposed to underwater explosions. You subject the materials to various magnitudes of explosions, then record whether or not the hull cracked. In the life sciences, a common experiment would be the bioassay, where you subject organisms to various levels of a stress and record whether or not they survive.

Probit analysis can answer these kinds of questions: For each hull material, what shock level cracks 10% of the hulls? What concentration of a pollutant kills 50% of the fish? Or, at a given pesticide application, what is the probability that an insect dies?

bit AnalysisUse probit analysis when you want to estimate percentiles, survival probabilities, and cumulative probabilities for the distribution of a stress, and draw probability plots. When you enter a factor and choose a Weibull, lognormal, or loglogistic distribution, you can also compare the potency of the stress under different conditions.

MINITAB calculates the model coefficients using a modified Newton-Raphson algorithm.

Data

Enter the following columns in the worksheet:

n two columns containing the response variable, set up in success/trial or response/frequency format

n one column containing a stress variable (treated as a covariate in MINITAB)

n (optional) one column containing a factor

Probit Analysis Probit Analysis

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Response variable

The response data is binomial, so you have two possible outcomes, success or failure. You can enter the data in either success/trial or response/frequency format. Here is the same data arranged both ways:

Temp80120140160

Succes



Factors

Text categories (factor levels) are processed in alphabetical order by default. If you wish, you can define your own ordersee Ordering Text Categories in the Manipulating Data chapter of MINITAB Users Guide 1 for details.

h To perform a probit analysis

How you run the analysis depend on whether your worksheet is in success/trial or response/frequency format.

1 Choose Stat Reliability/Survival Probit Analysis.

Response Frequency Temp1 2 800 8 801 4 1200 6 1201 7 1400 3 1401 9 1600 1 160

Response/frequency format

The Success column contains the number of successes; the Trials column contains the number of trials.

The Response column contains values which indicate whether the unit succeeded or failed. The higher value corresponds to a success. The Frequency column indicates how many times that observation occurred.

Success Trials2 104 107 109 10

s/trial format

Chapter 17 Probit Analysis

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2 Do one of the following:

n Choose Responses in success/trial format.1 In Number of successes, enter one column of successes.



2 In Number of trials, enter one column of trials.

n Choose Responses in response/frequency format.1 In Response, enter one column of response values.

2 If you have a frequency column, enter the column in with frequency.

3 In Stress (stimulus), enter one column of stress or stimulus levels.

4 If you like, use any of the options described below, then click OK.

Options

Probit Analysis dialog box

n include a factor in the modelsee Probit Analysis on page 17-2.

n choose one of seven common lifetime distributions, including the normal (default), lognormal basee, lognormal base10, logistic, loglogistic, Weibull, and extreme value distributions.

Estimate subdialog box

n estimate percentiles for the percents you specifysee Percentiles on page 17-8. These percentiles are added to the default table of percentiles.

n estimate survival probabilities for the stress values you specifysee Survival and cumulative probabilities on page 17-9.

n specify fiducial (default) or normal approximation confidence intervals.

n specify a confidence level for all of the confidence intervals. The default is 95%.

Graphs subdialog box

n suppress the display of the probability plot.

n draw a survival plotsee Survival plots on page 17-10.

n do not include confidence intervals on the above plots.

n plot the Pearson or deviance residuals versus the event probability. Use these plots to identify poorly fit observations.


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Options subdialog box

n enter starting values for model parameterssee Estimating the model parameters on page 17-11.



n change the maximum number of iterations for reaching convergence (the default is 20). MINITAB obtains maximum likelihood estimates through an iterative process. If the maximum number of iterations is reached before convergence, the command terminatessee Estimating the model parameters on page 17-11.

n use historical estimates for the model parameters. In this case, no estimation is done; all resultssuch as the percentilesare based on these historical estimates. See Estimating the model parameters on page 17-11.

n estimate the natural response rate from the data or specify a valuesee Natural response rate on page 17-12.

n if you have response/frequency data, you can define the value used to signify the occurrence of a success. Otherwise, the highest value in the column is used.

n enter a reference level for the factorsee Factor variables and reference levels on page 17-11. Otherwise, the lowest value in the column is used.

n perform a Hosmer-Lemeshow test to assess how well your model fits the data. This test bins the data into 10 groups by default; if you like, you can specify a different number.

Results subdialog box

n display the following in the Session window: no output the basic output, which includes the response information, regression table, test

for equal slopes, the log-likelihood, multiple degrees of freedom test, and two goodness-of-fit tests

the basic output, plus distribution parameter estimates and the table of percentiles and/or survival probabilities (default)

the above output, plus characteristics of the distribution and the Hosmer-Lemeshow goodness-of-fit test

n show the log-likelihood for each iteration of the algorithm.

Note When you select fiducial confidence intervals, MINITAB will display fiducial confidence intervals for the median, Q1, and Q2 and normal confidence intervals for mean, standard deviation, and IQR in the characteristics of distribution table.


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Storage subdialog box

n store the Pearson and deviance residuals

n store the characteristics of the fitted distribution, including



percentiles and their percents, standard errors, and confidence limits survival probabilities and their stress level and confidence limits

n store information on the estimated equation, including event probability estimated coefficients and standard error of the estimates variance/covariance matrix natural response rate and standard error of the natural response log-likelihood for the last iteration

Output

The default output consists of:

n the response information

n the factor information

n the regression table, which includes the estimated coefficients and their standard errors. Z-values and p-values. The Z-test tests that the coefficient is significantly different

than 0; in other words, is it a significant predictor? natural response ratethe probability that a unit fails without being exposed to

any of the stress.

n the test for equal slopes, which tests that the slopes associated with the factor levels are significantly different.

n the log-likelihood from the last iteration of the algorithm.

n two goodness-of-fit tests, which evaluate how well the model fits the data. The null hypothesis is that the model fits the data adequately. Therefore, the higher the p-value the better the model fits the data.

n the parameter estimates for the distribution and their standard errors and 95% confidence intervals. The parameter estimates are transformations of the estimated coefficients in the regression table.

n the table of percentiles, which includes the estimated percentiles, standard errors, and 95% fiducial confidence intervals.

n the probability plot, which helps you to assess whether the chosen distribution fits your datasee Probability plots on page 17-10.


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n the relative potencycompares the potency of a stress for two levels of a factor. To get this output, you must have a factor, and choose a Weibull, lognormal, or loglogistic distribution.

Suppose you want to compare how the amount of voltage affects two types of light



bulbs, and the relative potency is .98. This means that light bulb 1 running at 117 volts would fail at approximately the same time as light bulb 2 running at 114.66 volts (117 .98).

Probit model and distribution function

MINITAB provides three main distributionsnormal, logistic, and extreme valueallowing you to fit a broad class of binary response models. You can take the log of the stress to get the lognormal, loglogistic, and Weibull distributions, respectively. This class of models (for the situation with no factor) is defined by:

pij =

where

The distribution functions are outlined below:

Here, pi in the Variance column of the table is 3.14159.

The distribution function you choose should depend on your data. You want to choose a distribution function that results in a good fit to your data. Goodness-of-fit statistics can be used to compare fits using different distributions. Certain distributions may be used for historical reasons or because they have a special meaning in a discipline.

pij = the probability of a response for the jth stress level

g(yj) = the distribution function (described below)

0 = the constant termxj = the j

th level of the stress variable

= unknown coefficient associated with the stress variablec = natural response rate

Distribution Distribution function Mean Variance

logistic g(yj) = 0 pi2 / 3

normal g(yj) = (yj) 0 1

extreme value g(yj) = (Euler constant) pi2 / 6

c 1 c( )g 0 xj+ ( )+

1 1 e yj+( )

1 e eyj


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Percentiles

At what stress level do half of the units fail? How much pesticide do you need to apply to kill 90% of the insects? You are looking for percentiles.



Common percentiles used are the 10th, 50th, and 90th percentiles, also known in the life sciences as the ED 10, ED 50 and ED 90 (ED = effective dose).

The probit analysis automatically displays a table of percentiles in the Session window, along with 95% fiducial confidence intervals. You can also request:

n additional percentiles to be added to the table

n normal approximation rather than fiducial confidence intervals

n a confidence level other than 95%

The Percentile column contains the stress level required for the corresponding percent of the events to occur.

In this example, you exposed light bulbs to various voltages and recorded whether or not the bulb burned out before 800 hours.

h To modify the table of percentiles

1 In the Probit Analysis main dialog box, click Estimate.

2 Do any of the following:

n In Estimate percentiles for these additional percents, enter the percents or a column of percents.

At 104.9931 volts, 1% of the bulbs burn out before 800 hours.

Table of Percentiles Standard 95.0% Fiducial CIPercent Percentile Error Lower Upper 1 104.9931 1.3715 101.9273 107.3982 2 106.9313 1.2661 104.1104 109.1598 3 108.1795 1.1997 105.5144 110.2980 4 109.1281 1.1504 106.5795 111.1656

etc.


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n Choose Normal approximation to request normal approximation rather than fiducial confidence intervals.

n Change the confidence level for the percentiles (default is 95%): In Confidence level, enter a value. This changes the confidence level for all confidence intervals.



Survival and cumulative probabilities

What is the probability that a submarine hull will survive a given strength of shock? At a given pesticide application, what is the probability that an insect survives? You are looking for survival probabilitiesestimates of the proportion of units that survive at a certain stress level.

When you request survival probabilities, they are displayed in a table in the Session window. In this example, we exposed light bulbs to various voltages and recorded whether or not the bulb burned out before 800 hours. Then we requested a survival probability for light bulbs subjected to 117 volts:

To calculate cumulative probabilities (the likelihood of failing rather than surviving), subtract the survival probability from 1. In this case, the probability of failing before 800 hours at 117 volts is 0.2308.

h To request survival probabilities

1 In the Probit Analysis main dialog box, click Estimate.

2 In Estimate survival probabilities for these stress values, enter one or more stress values or columns of stress values.

Table of Survival Probabilities 95.0% Fiducial CI Stress Probability Lower Upper 117.0000 0.7692 0.6224 0.8825

The probability of a bulb lasting past 800 hours is 0.7692 at 117 volts.


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Probability plots

A probability plot displays the percentiles. You can use the probability plot to assess whether a particular distribution fits your data. In general, the closer the points fall to



the fitted line, the better the fit.

For a discussion of probability plots, see Probability plots on page 15-37.

When you have more than one factor level, lines and confidence intervals are drawn for each level. If the plot looks cluttered, you can turn off the confidence intervals in the Graphs subdialog box. You can also change the confidence level for the 95% confidence by entering a new value in the Estimate subdialog box.

Survival plots

Survival plots display the survival probabilities versus stress. Each point on the plot represents the proportion of units surviving at a stress level. The survival curve is surrounded by two outer linesthe 95% confidence interval for the curve, which provide reasonable values for the true survival function.

For an illustration of a survival plot, see Survival plots on page 15-40.

h To draw a survival plot

1 In the Probit Analysis dialog box, click Graphs.

2 Check Survival plot.

3 If you like, turn off the 95% confidence intervaluncheck Display confidence intervals on above plots. Click OK.

4 If you like, change the confidence level for the 95% confidence intervalclick Estimate. In Confidence level, enter a value. Click OK.


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Factor variables and reference levels

You can enter numeric, text, or date/time factor levels. MINITAB needs to assign one factor level to be the reference level, meaning that the estimated coefficients are



interpreted relative to this level.

Probit analysis creates a set of design variables for the factor in the model. If there are k levels, there will be k-1 design variables and the reference level will be coded with all 0s. Here are two examples of the default coding scheme:

By default, MINITAB designates the lowest numeric, date/time, or text value as the reference factor level. If you like, you can change this reference value in the Options subdialog box.

Estimating the model parameters

MINITAB uses a modified Newton-Raphson algorithm to estimate the model parameters. If you like, you can enter historical estimates for these parameters. In this case, no estimation is done; all resultssuch as the percentilesare based on these historical estimates.

When you let MINITAB estimate the parameters from the data, you can optionally:

n enter starting values for the algorithm.

n change the maximum number of iterations for reaching convergence (the default is 20). MINITAB obtains maximum likelihood estimates through an iterative process. If the maximum number of iterations is reached before convergence, the command terminates.

Why enter starting values for the algorithm? The maximum likelihood solution may not converge if the starting estimates are not in the neighborhood of the true solution, so you may want to specify what you think are good starting values for parameter estimates.

Factor A with 4 levels(1 2 3 4)

Factor B with 3 levels(High Low Medium)

reference level

A1 A2 A3 reference level

B1 B21 0 0 0 High 0 02 1 0 0 Low 1 03 0 1 0 Medium 0 14 0 0 1


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h To control estimation of the parameters

1 In the Probit Analysis main dialog box, click Options.



2 Do one of the following:

n To estimate the model parameters from the data (the default), choose Estimate model parameters. To enter starting estimates for the parameters: In Use starting estimates, enter

one starting value for each coefficient in the regression table. Enter the values in the order that they appear in the regression table.

To specify the maximum number of iterations, enter a positive integer.

n To enter your own estimates for the model parameters, choose Use historical estimates and enter one starting value for each coefficient in the regression table. Enter the values in the order that they appear in the regression table.

Natural response rate

The regression table includes the natural response ratethe probability that a unit fails without being exposed to any of the stress. This statistic is used in situations with high mortality or high failure rates. For example, you might want to know the probability that a young fish dies without being exposed to a certain pollutant. If the natural response rate is greater than 0, you may want to consider the fact that the stress does not cause all of the deaths in the analysis.

You can choose to estimate the natural response rate from the data, or set the value. You would set the value when you have a historical estimate, or to use as a starting value for the algorithm.

Note Do not enter a starting value for the natural response rate here.


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e Example of a probit analysis

Suppose you work for a lightbulb manufacturer and have been asked to determine bulb life for two types of bulbs at typical household voltages. The typical line voltage entering a house is 117 volts + 10% (or 105 to 129 volts).



You subject the two bulbs to five stress levels within that range108, 114, 120, 126, and 132 volts, and define a success as: The bulb fails before 800 hours.

1 Open the worksheet LIGHTBUL.MTW.

2 Choose Stat Reliability/Survival Probit Analysis.

3 Choose Response in success/trial format.

4 In Number of successes, enter Blows. In Number of trials, enter Trials.

5 In Stress (stimulus), enter Volts.

6 In Factor (optional), enter Type. In Enter number of levels, enter 2.

7 From Assumed distribution, choose Weibull.

8 Click Estimate. In Estimate survival probabilities for these stress values, enter 117. Click OK.

9 Click Graphs. Uncheck Display confidence intervals on above plots. Click OK in each dialog box.

Sessionwindowoutput

Probit Analysis: Blows, Trials versus Volts, Type Distribution: Weibull

Response Information

Variable Value CountBlows Success 192 Failure 308Trials Total 500

Factor Information

Factor Levels Values Type 2 A B

Estimation Method: Maximum Likelihood


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Regression Table StandardVariable Coef Error Z PConstant -97.019 7.673 -12.64 0.000Volts 20.019 1.587 12.61 0.000



Type B 0.1794 0.1598 1.12 0.262NaturalResponse 0.000

Test for equal slopes: Chi-Square = 0.2585, DF = 1, P-Value = 0.611Log-Likelihood = -214.213

Goodness-of-Fit Tests

Method Chi-Square DF PPearson 2.516 7 0.926Deviance 2.492 7 0.928

Type = A

Tolerance Distribution

Parameter Estimates Standard 95.0% Normal CIParameter Estimate Error Lower UpperShape 20.019 1.587 17.138 23.384Scale 127.269 0.737 125.832 128.722

Table of Percentiles Standard 95.0% Fiducial CI Percent Percentile Error Lower Upper 1 101.1409 1.8424 96.9868 104.3407 2 104.7307 1.6355 101.0429 107.5731 3 106.9008 1.5090 103.5009 109.5267 4 108.4760 1.4171 105.2866 110.9457 5 109.7203 1.3449 106.6975 112.0680 6 110.7531 1.2854 107.8683 113.0007 7 111.6387 1.2348 108.8717 113.8017 8 112.4158 1.1909 109.7516 114.5057 9 113.1096 1.1523 110.5364 115.1354 10 113.7373 1.1177 111.2458 115.7062 20 118.0817 0.8986 116.1208 119.7003 30 120.8808 0.7901 119.2012 122.3424 40 123.0693 0.7358 121.5505 124.4720 50 124.9600 0.7179 123.5231 126.3718 -----the rest of this table omitted for space-----


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Type = B

Tolerance Distribution

Parameter Estimates Standard 95.0% Normal CIParameter Estimate Error Lower UpperShape 20.019 1.587 17.138 23.384Scale 126.134 0.704 124.761 127.522

Table of Percentiles Standard 95.0% Fiducial CI Percent Percentile Error Lower Upper 1 100.2388 1.8617 96.0399 103.4706 2 103.7965 1.6562 100.0595 106.6728 3 105.9472 1.5303 102.4960 108.6073 4 107.5084 1.4386 104.2667 110.0121 5 108.7416 1.3663 105.6661 111.1226 6 109.7652 1.3065 106.8277 112.0453 7 110.6429 1.2556 107.8234 112.8374 8 111.4131 1.2113 108.6967 113.5335 9 112.1007 1.1722 109.4760 114.1558 10 112.7228 1.1371 110.1805 114.7197 20 117.0285 0.9108 115.0289 118.6590 30 119.8026 0.7929 118.1018 121.2561 40 121.9716 0.7280 120.4520 123.3436 50 123.8454 0.6989 122.4294 125.2031

-----the rest of this table omitted for space-----


Table of Relative Potency

Factor: Type Relative 95.0% Fiducial CIComparison Potency Lower UpperA VS B 0.9911 0.9754 1.0068

Chapter 17 References

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Ref

Graphwindowoutput



Interpreting the results

The goodness-of-fit tests (p-values = 0.926, 0.928) and the probability plot suggest that the Weibull distribution fits the data adequately. Since the test for equal slopes is not significant (p-value = .611), the comparison of lightbulbs will be similar regardless of the voltage level. In this case, the lightbulbs A and B are not significantly different because the coefficient associated with type B is not significantly different than 0 (p-value = .262).

At 117 volts, what percentage of the bulbs last beyond 800 hours? Eight-three percent of the bulb As and 80% of the bulb Bs last beyond 800 hours.

At what voltage do 50% of the bulbs fail before 800 hours? The table of percentiles shows you that 50% of bulb As fail before 800 hours at 124.96 volts; 50% of bulb Bs fail before 800 hours at 123.85 volts.

erences[1] D.J. Finney (1971). Probit Analysis, Cambridge University Press.

[2] D.W. Hosmer and S. Lemeshow (1989). Applied Logistic Regression, John Wiley & Sons, Inc.

[3] P. McCullagh and J.A. Nelder (1992). Generalized Linear Models, Chapman & Hall.

[4] W. Murray, Ed. (1972). Numerical Methods for Unconstrained Optimization, Academic Press.

[5] W. Nelson (1982). Applied Life Data Analysis, John Wiley & Sons.

Probit Analysis OverviewProbit AnalysisDataOptionsOutputProbit model and distribution functionPercentilesSurvival and cumulative probabilitiesProbability plotsSurvival plotsFactor variables and reference levelsEstimating the model parametersNatural response rate

References

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17 Probit Analysis

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