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[null,null,"Crash Modification Factor Regression Software"]
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[null,null,null,null,null,null,"Build 03.5"]
[null,"FOREWORD"]
[null,"This software can be used to conduct a statistical analysis of crash modification factors (CMFs) for a common treatment or change in site characteristic."]
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[null,"The \"Introduction\" worksheet provides useful information about this software.  This information should be read prior to first use of the software.  The Introduction worksheet can be selected from the tabs at the bottom of the screen.   After reading the Introduction, the other worksheets can be used to analyze CMFs."]
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[null,"This software is intended for use with the document identified below.  The analyst is encouraged to read this document so that he or she will have an understanding of how best to use the software and interpret its output."]
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[null,"Carter, D., R. Srinivasan, F. Gross, S. Himes, T. Le, B. Persaud, C. Lyon, and J. Bonneson. Guidance for the Development and Application of Crash Modification Factors – Final Report. NCHRP Project 17-63. University of North Carolina, North Carolina, 2017."]
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[null,"    Appendix A: Procedure for Estimating the Effect of a Proposed Treatment at a Site of Interest."]
[null,"    Appendix B: Procedure for Disaggregating Aggregate CMFs."]
[null,"    Appendix C: Homogeneity Test and Procedure for Combining CMFs."]
[null,"    Appendix N: User Guide for CMF Regression Software."]
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[null,"ACKNOWLEDGMENT OF SPONSORSHIP"]
[null,"This work was sponsored by the American Association of State Highway and Transportation Officials (AASHTO), in cooperation with the Federal Highway Administration, and was conducted in the National Cooperative Highway Research Program (NCHRP), which is administered by the Transportation Research Board of the National Academies.  "]
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[null,"DISCLAIMER"]
[null,"This spreadsheet software is offered as is, without warranty or promise of support of any kind either expressed or implied. Under no circumstance will the National Academy of Sciences or the Transportation Research Board (TRB) be liable for any loss or damage caused by the installation or use of this product. The National Academy of Sciences, TRB, and the TRB National Cooperative Highway Research Program (NCHRP) make no representation or warranty of any kind, expressed or implied, in fact or in law, including without limitations the warranty of merchantability or the warranty of fitness for a particular purpose, and shall not in any case be liable for any consequential or special damages. "]
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[null,"This computational engine is subject to change without notice and does not represent a commitment on the part of the National Academy of Sciences, the TRB, or the NCHRP to notify any person of such revisions."]
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[null,"COPYRIGHT © 2015"]

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[null,null,"INTRODUCTION"]
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[null,"PREFACE"]
[null,"This software package can be used to evaluate two or more CMFs associated with a common treatment. The evaluation can consist of: (1) computing the overall average CMF, (2) computing CMFs by crash type or severity category using reported aggregate CMFs, or (3) computing CMFs as a function of site characteristics."]
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[null,"The evaluation is based on a weighted statistical analyses, where the reported standard error of the CMF is used to compute its weight. Statistical tests are implemented to assist the analyst in determining whether the CMFs are more appropriately described using an overall average value or  a function of site characteristics.  Weighted regression is used to quantify the regression coefficients."]
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[null,"EVALUATION SCOPE"]
[null,"The methods used in this software are based on the premise that CMF values (and their respective standard errors) for two or more locations are available from the literature and that the values are unbiased.  They are also based on the premise that more detailed data (e.g., crash counts) are not available for the collective set of locations. There are more robust procedures available for computing an overall average CMF when detailed data (e.g., crash data) are available for the collective set of similar locations (e.g., Hauer, 1997). These procedures should be used to compute the overall average CMF when detailed data are available."]
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[null,"High quality CMFs should be used in this software. Sources of high quality CMFs include the HSM and the FHWA CMF Clearinghouse. Criteria for determining the quality of a CMF are described in the Introduction to Part D of the HSM. The Clearinghouse uses a star quality rating system that is described on its website (http://www.cmfclearinghouse.org/sqr.cfm)."]
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[null,"WORKSHEET SELECTION"]
[null,"This software includes the following worksheets:"]
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[null,"● Welcome ",null,null,"Includes a foreword, acknowledgements, and disclaimer."]
[null,"● Introduction",null,null,"Brief overview of the software and guidelines for its use."]
[null,"● Set Up",null,null,"Input data to describe the model structure and database."]
[null,"● Main ",null,null,"Input CMF data and start calculations."]
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[null,">>> Always proceed to the Set Up worksheet to start the evaluation process. <<<"]
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[null,"USER'S GUIDE OVERVIEW"]
[null,"This section provides a brief overview of the information needed to use this software. The complete User Guide for this software is provided as Appendix N to the NCHRP Project 17-63 Final Report (as identified in the Welcome worksheet)."]
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[null,"Each worksheet is designed in a consistent manner and their use is similar.  The cells with a blue background are for user input.   Most of the other cells are locked to prevent inadvertent changes to cell contents.  A sample portion of the Set Up worksheet is shown below. "]
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[null,"Set Up Worksheet"]
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[null,null,null,null,null,null,"Place cursor over the red corner to obtain"]
[null,null,null,null,null,null,"a pop-up message with useful information."]
[null,"General Information"]
[null,"The analyst can enter any descriptive information about the location being analyzed.  Use the blue cells to enter this data."]
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[null,"Model Description"]
[null,"The model structure to be used is entered. Two choices are currently supported: \"Model 1\" and \"User Defined.\" The structure of Model 1 is described in equation form in this section of the worksheet. If selected, the software will enter the equation in the column headed \"Predicted CMF\" in the Main worksheet.  It will use this equation to compute the predicted CMF for each crash distribution category."]
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[null,"If \"User Defined\" is selected, the analyst will need to enter the equation in the column headed \"Predicted CMF\" in the Main worksheet. The equation will need to be entered using Excel formula protocols. It will need to be copied to each row for which there is an observation.  The cells in which an equation is to be entered will have a grey background."]
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[null,"The number of variables used to describe the crash distribution is entered in the blue cell provided.  This number must have a value of one or larger.  The crash distribution variables can describe the crash severity distribution, the crash type distribution, or a combination of both crash severity and type. "]
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[null,"The number of site-characteristic variables is entered in the blue cell provided. This number must have a value of zero or larger.  The analyst can include any site characteristics that he/she believes may influence the value of the CMF.  Example site characteristics include: area type (i.e, urban/rural), traffic volume, lane width, etc.  "]
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[null,"If a site characteristic has \"n\" levels, then it can be represented as a series of \"n-1\" indicator variables.  For example, if the variable describes area type and it has 2 levels (i.e., urban or rural), then one (=2-1) site characteristic variable (e.g., urban) is represented in the data.  This variable has a value of 1.0 if the site is urban and 0.0 if the site is not urban.  "]
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[null,"A site characteristic variable can be entered into the Main worksheet using its original units, or it can be transformed to alternative units if desired. For example, if an analysis of the data indicates that the reported CMFs vary with the natural log of the AADT, then the natural log of the AADT should be entered in the Main worksheet as a site characteristic variable."]
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[null,"Data Description"]
[null,"The number of CMF observations is entered in the blue cell provided.  The minimum number of observations must exceed by 2 the sum of the crash distribution and site characteristic variables.   The spreadsheet can be used to evaluate up to 500 observations."]
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[null,"Main Worksheet"]
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[null,"The reported CMF data are entered in the blue cells shown in the figure above.  Sample data are shown for four observations (note: these data can be cleared by clicking on the Clear Data button, described below).  The first column is used to enter text to describe each observation (e.g., \"Obs 1\", \"Obs 2\", etc.).  The first blue row is used to enter text to describe each variable.  For example, the data in the column headed \"p1\" are labeled \"p_mv_fi\" indicating the column contains data describing the proportion of multiple-vehicle fatal-and-injury crashes."]
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[null,"The reported CMF values are entered in the blue cells in the column headed \"CMF\". The standard error associated with each reported CMF is entered in the column headed \"Std Err\".  "]
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[null,"If data are to be entered by copying data from another spreadsheet into the blue cells of the Main worksheet, then paste by \"value\" only (i.e., Ctrl-Alt-V, Value, OK).  Do not use a simple paste, or similar, because it will delete the cell background color and borders. The software relies on the correct cell background color to work properly."]
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[null,"Crash Distribution Variables"]
[null,"The  figure above shows four columns headed \"p1\", \"p2\", \"p3\", and \"p4\".  These columns correspond to the four crash distribution variables that were indicated in the data entry of the Set Up worksheet.  In general, the number of columns headed \"p1\", ..., \"pn\" will be equal to the number of crash distribution variables (n) indicated in the data entry of the Set Up worksheet.  For a given observation (i.e., row), the crash distribution proportions should add to 1.000."]
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[null,"Site Characteristic Variables"]
[null,"The figure above shows three columns headed \"c1\", \"c2\", and \"c3\". These columns corresponde to the three site characteristic variables indicated in the data entry of the Set Up worksheet. In general, the number of columns headed \"c1\", ..., \"cm\" will be equal to the number of site characteristic variables (m) indicated in the data entry of the Set Up worksheet."]
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[null,"The top row of the figure above is labled \"Include:\".  The analyst can optionally include or exclude site characteristic variables from the analysis.  To include a variable, the analyst will need to enter \"Yes\" in the blue cell in the associated column.  To exclude a variable, the analyst will need to enter \"No.\"  If a variable is excluded, the cell background and text color will be changed to white. The data will NOT be deleted but it will be excluded from the analysis.  To restore the data in the analysis, change a \"No\" to a \"Yes\".  "]
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[null,"Run Control"]
[null,"Two grey buttons are provided to control the evaluation. These buttons are shown in the figure above.  The analyst clicks on a button to effect the indicated operation.  One button is labeled \"Clear Data\". This button is used to erase the data in the blue cells.  It can be used to clear old data from the spreadsheet, before entering new data."]
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[null,"The button labeled \"Analyze Data\" is used after the data are entered in the blue cells provided in the Main worksheet. When this button is clicked, the regression analysis is conducted and the results are shown in rows 13 to 17. Various diagnostic messages are provided in rows 9 to 13."]
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[null,"Warning Mesages"]
[null,"When the \"Analyze Data\" button is clicked, the data are checked to ensure the values are appropriate.  In particular, the blue cells are checked to ensure that values are entered (i.e., no cells are blank).  If a blank cell is found, then the analysis is stopped and a warning message provided in the section labeled \"Warning Messages\". An exception is the blue cells used to label the columns or rows.  The label cells can be left blank. "]
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[null,"The entered CMFs and associated standard errors are checked to ensure that there are no values that are not positive (i.e., zero or negative).  The analysis is stopped and a warning message is provided if one or more values are not positive."]
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[null,"If the model structure is indicated as \"User Defined\" in the Set Up worksheet, then the analyst must enter the desired equation in the cells with grey background.  If these equations are not entered, then the analysis is stopped and a warning message is provided."]
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[null,"Excel Solver is used to compute the regression coefficients. If Solver is unable to converge on a reliable set of coefficients then a message is provided and the analysis is stopped."]
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[null,"Diagnostic Messages"]
[null,"Several diagnostic messages are provided to help the analyst interpret the analysis results.  These messages are provided in the section labeled \"Diagnostic Messages\" of the Main worksheet.  Example messages are shown in the figure above."]
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[null,"The first message reports the percentage of observations with a weight less than 4.0. CMFs with a smaller weight are less reliable because they are based on a relatively small sample size (i.e., a small number of observed crashes). A majority (desirably all) of the CMFs should have a weight of 4.0 or more. "]
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[null,"Excel Solver is used to compute the regression coefficients. If Solver is able to converge on a reliable set of coefficients then a message is provided and the computed CMFs are displayed."]
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[null,"The third diagnostic message describes the results from the Chi-Square test of the treatment effect. The null hypothesis is that the treatment does not have an effect on safety. The statistics associated with this test are shown in the second-to-last row of the figure above.  A significance level (alpha) of 0.05 is used to determine whether to reject the null hypothesis."]
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[null,"The fourth diagnostic message describes the results from the Chi-Square test of homogenity in the CMF values. The null hypothesis is that predicted CMF equals the observed CMF. The statistics associated with this test are shown in the last row of the figure above.  A significance level (alpha) of 0.05 is used to determine whether to reject the null hypothesis."]
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[null,"The standard error of the regression coefficients is computed using the Jacobian matrix of first derivatives of the regression model coefficients. If this matrix is not positive definite, then the Jacobian matrix cannot be used.  In this situation, the standard errors are estimated using the Jackknife technique.  The need to use the Jackknife technique is indicated by a fifth diagnostic message (if this message is not presented, then the Jacobian matrix is used to estimate the standard error)."]
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[null,"Standardized Residual"]
[null,"The figure above is produced in the Main worksheet. It displays the standardized residual for each observation.  A standardized residual larger than +3.0 or smaller than -3.0 may be an outlier (i.e., input data may have an error, or observation may from a site with unexplained attributes that are different from those of the other sites).  The standardized residual (r) is computed as r = (observed CMF - predicted CMF)/standard error."]
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[null,"Regression Coefficients"]
[null,"After the \"Analyze Data\" button is clicked, the best-fit regression coefficients are shown in the Main worksheet. The are shown in the second-to-last row of the figure above. These coefficients correspond to variables \"p1\", ... \"pn\", and \"c1\", ..., \"cm\", as defined in the Set Up worksheet.  These coefficients are used in the equations in the column headed \"Predicted CMF\" to compute the predicted CMF for each observation."]
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[null,"The standard error of each regression coefficient is provided in the row below the corresponding regression coefficient. The standard error is used to estimate the variability in the coefficient values.  Coefficients with a relatively large standard error are less confidently known than those coefficients with a small standard error.  If a site characteristic variable is not known with the desired level of confidence, it can be excluded from the regression model by entering \"No\" in the \"Include\" row of the Main worksheet."]
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[null,"Average CMF"]
[null,"After the \"Analyze Data\" button is clicked, the best-fit regression coefficients are used to compute the average CMF for each crash distribution category.  The average CMFs are shown in the second row of the figure above.  If one or more site characteristic variables are included in the model, then the computed average CMF includes the effect of the site characteristics at the values indicated in the \"Enter Value\" row.  "]
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[null,"In the figure above, the analyst has entered \"1\", \"1\", and \"0\" for the three site characteristic variables \"c1\", \"c2\", and \"c3\", respectively.  The variable \"c1\" is an indicator variable indicating the state whose data were used to quantify the CMF (= 1 for Minnesota or Missouri, 0 for Pennsylvania).  The variable \"c2\" is an indicator variable for roadway class (= 1 for freeway, 0 for non-freeway).  The variable \"c3\" is for roadway class (= 1 for multilane highway, 0 for non-multilane highway).  Based on the three entered values shown, the four average CMFs listed in the figure above correspond to freeways in Minnesota or Missouri.  The analyst can enter other values in the \"Enter Values\" row (and click on the Analyze Data button) to compute the average CMFs for other combinations of site characteristics."]
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[null,"In the figure above, the average CMF in the column headed \"p1\" is 1.004.  It was noted in a previous figure that this column describes the proportion of multiple-vehicle fatal-and-injury crashes.  So, the average CMF for multiple-vehicle fatal-and-injury crashes on freeways in Minnesota or Missouri is 1.004."]
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[null,"The standard error of the computed average CMF is also provided in the Main worksheet. This statistic is shown in the third row of the figure above.  It is computed using the variance-covariance matrix of the regression model.   The standard error of the CMF describes the distribution of the average CMF when the treatment is applied to similar sites.  It is used to estimate the 95th percentile confidence interval of the average CMF.  This confidence interval is shown in the fourth row of the figure above.  The confidence interval is based on an assumed lognormal distribution of the average CMF. "]
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[null,"In the figure above, the standard error of the average CMF in the column headed \"p1\" is 0.125.  It was noted previously that this column describes the proportion of multiple-vehicle fatal-and-injury crashes. So, this statistic describes the standard error of the average CMF (i.e., 1.004) for freeways in Minnesota or Missouri. The 95th percentile confidence interval is 0.78 to 1.27."]
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[null,"If the analyst selects \"User Defined\" for the model form, then the average CMF and standard error cannot be computed because the derivatives of a user defined model form are unknown."]
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[null,"Application Process"]
[null,"The following steps describe the use of this spreadsheet to conduct a regression analysis of CMFs."]
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[null,"Step 1. Describe the Model and Database in the Set Up Worksheet"]
[null,"The analyst goes first to the Set Up worksheet to describe the model and the data.  "]
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[null,"Step 2.  Enter the Data in the Main Worksheet"]
[null,"The analyst clicks on the \"Clear Data\" button to clear all data remanents from prior analyses. Then, the data are entered in the blue cells.  These data include the reported CMFs and their standard errors, as well as the crash distribution data (if disaggregate CMFs are sought) and the site characteristrics (if CMF functions are sought)."]
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[null,"If the analyst has chosen \"User Defined\" model in the Set Up worksheet, then the equation for the model is entered in each of the cells with a grey background.  The desired equation must use Excel formula protocols.  This equation must then be copied down for all rows with observations."]
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[null,"Step 3. Enter Values of the Site Characteristic Variables"]
[null,"If site characteristic variables are used and if the model form is not User Defined, then the site characteristics for which the average CMF will be computed are entered in the blue cells on the \"Enter Value\" row. That is, if in the Set Up worksheet a pre-defined model is selected and it includes one or more site characteristic variables, then the analyst should enter values for the site characteristics for which an estimate of the average CMF  is desired."]
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[null,"If no site characteristics are used or, if the model form is User Defined, then values do not need to be entered in the blue cells on the \"Enter Value\" row.  This step can be skipped."]
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[null,"Step 4. Initiate Calculations"]
[null,"Click on the \"Analyze Data\" button to initiate the calculation process.  Assess the validity of the observations using the standardized residuals.  Correct or eliminate the observations as needed. Assess the variablity of the regression coefficients using the standard error of the regression coefficients.  Include or exclude a site characteristic variable as needed.  Consider the diagnostic messages.  Determine if more (or less) site characteristic variables may be helpful."]
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[null,"Record the computed average CMFs for the specified site characteristic variables.  Consider also the standard deviation of these CMFs.If the variability is large, then the average CMF value may not be sufficiently well known to represent the basis for reliable investment decisions.  If the confidence interval includes 1.00, then there is a chance that the treatment may not improve safety at every site."]
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[null,"If it is desired to estimate CMF values for other combinations of site characteristics, then repeat Steps 3 and 4 using the new values for the site characteristic variables."]
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[null,"Overall Average CMF.  To compute an overall average CMF value using this spreadsheet, in the Set Up worksheet enter \"1\" for the number of crash distribution variables; enter \"0\" for the number of site characteristic variables.  In the Main worksheet, use the cells with blue background to enter \"1\" for each observation in the column headed \"p1\"."]
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[null,"The Solver Add-In must be installed before this spreadsheet will work.  For EXCEL 2007, select the Tools menu, choose Add-Ins.  For EXCEL 2013, select Developer, choose Add-Ins.  If Solver is listed, the check its check box.  If Solver is not listed, then click Browse, navigate to the folder where Solver.xlam is located, click on Solver.xlam, and click OK to close the Browse box. Click OK to close the Add-In box."]
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[null,"If Solver was just installed for the first time in Excel, it will need to be used at least once manually before it can be used by this spreadsheet. That is, Solver is not \"opened\" for use by macros until it has been used once manually. To manually use solver, apply the following steps: (1) select the Data menu (for Excel 2007, select the Tools menu).  (2) Select Solver. (4) Set the \"target cell\" to $F$49. (5) Click the \"Value of\" button and set the  \"Value of\" cell to 50. (6) Set the \"By Changing Cells\" to $F$48. (7) Click on Solve, and then click on OK.  The value of 7.071 should be showing in F48 and 50 should be showing in F49."]
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[null,null,null,null,"Changing cell:",2]
[null,null,null,null,"Target cell:",{"$f":"F345^2","v":4}]
[null,null,null,null,"Value of:",50]

{"sheet":"Set Up","index":2,"n_rows":37,"n_cols":12,"rows_kept":37,"cut":false,"hidden":false,"merged":[[1,1,1,11],[3,3,3,11],[4,3,4,4],[4,6,4,7],[27,6,27,7]]}
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[null,"Crash Modification Factor Regression Software"]
[null,"General Information"]
[null,"Project description:",null,"Sample Data"]
[null,"Analyst:",null,"JAB",null,"Date:",{"$f":"NOW()","v":"2016-11-21T18:59:46.380"}]
[null,"Model Description"]
[null,"Model 1",null,"CMF = [p1 x CMF1 + p2 x CMF2 + ... + pn x CMFn] x fa"]
[null,null,"with,"]
[null,null,null,"CMF1 = exp(b1)       CMF2 = exp(b2)      CMFn = exp(bn)"]
[null,null,null,"fa = exp(c1 x X1   +  c2 x X2 + ... +  cm x Xm)"]
[null,null,"where,"]
[null,null,null,"CMFi = CMF for crash distribution category i;"]
[null,null,null,"pi = proportion of crashes associated with crash category i (i = 1 to n);"]
[null,null,null,"bi = regression coefficient associated with crash distribution category i;"]
[null,null,null,"Xk = variable describing site characteristic k (k = 1 to m); and"]
[null,null,null,"ck = regression coefficient for site characteristic k."]
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[null,"Enter model structure to be used:",null,null,null,null,"Model 1"]
[null,"Number of variables in crash distribution (n):",null,null,null,null,4,{"$f":"IF(G29<1, \"Warning: Need at least one variable.\", \".\")","v":"."}]
[null,"Number of site characteristic variables (m):",null,null,null,null,3]
[null,"Data Description"]
[null,"Number of CMF observations:",null,null,null,null,36,{"$f":"IF(G32<2*(G29+G30), \"Warning: Reduce variables or increase observations to get reliable results.\", \".\")","v":"."}]
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[null,"Advanced Settings"]
[null,"Optimize scale parameter:",null,null,"Yes","Max. number of samples used to estimate standard error:",null,null,null,null,null,50]

{"sheet":"Main","index":3,"n_rows":56,"n_cols":32,"rows_kept":56,"cut":false,"hidden":false,"merged":[[1,16,4,20]]}
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[null,"Run Control",null,null,null,null,null,null,null,null,null,null,null,null,null,null,"A standardized residual larger than +3.0 or smaller than -3.0 may be an outlier (i.e., input data may have an error, or observation may from a site with unexplained attributes that are different from those of the other sites)."]
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[null,"Warning Messages"]
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[null,"Diagnostic Messages"]
[null,"8 percent of the observations have a weight less than 4.0."]
[null,"Solver found a solution. All constraints and optimality conditions are satisfied.",null,null,null,null,null,null,null,null,null,null,null,null,null,null,{"$f":"EXP(M16)*EXP(SUMPRODUCT($Q$11:$S$11,$Q$16:$S$16))","v":1.0043405608329459},{"$f":"EXP(N16)*EXP(SUMPRODUCT($Q$11:$S$11,$Q$16:$S$16))","v":0.88470939939511117}]
[null,"Treatment is likely to have some effect on crash frequency (reject Ho: no effect).",null,null,null,null,null,null,null,null,null,"Enter Value:",null,null,null,null,1,1,0]
[null,"The model accounts for the variation present (do not reject Ho: predicted CMF = observed CMF).",null,null,null,null,null,null,null,null,null,"Average CMF:",1.0043405608329459,0.88470939939511117,1.264424432361626,1.0940412607773851]
[null,null,null,null,null,null,null,null,null,null,null,"Std. Err. of CMF:",0.1249401518567681,8.8005707119140172E-2,9.537667299574247E-2,7.905539656335589E-2]
[null,null,null,null,null,null,null,null,null,null,null,"95% Conf. Interval:","0.78-1.27","0.72-1.07","1.09-1.46","0.95-1.26"]
[null,"Source","Chi-Squ.","d.f.","Prob.",null,"Sum LL",null,null,null,null,"Phi","p1","p2","p3","p4","c1","c2","c3"]
[null,"Treatment:",47.367725866724129,8,1.304691153707632E-7,null,{"$f":"SUM(G21:G56)","v":19.775768415897247},null,null,null,"Regression coefficients:",1.0269166620825778,-0.11987105552113479,-0.24669827345318618,0.11041480116681603,-3.4323804483663782E-2,0.11139623984994533,1.2805983440928979E-2,0.13571143276984263]
[null,"Homogeneity:",37.595449403410846,28,0.10630009621655979,null,null,null,null,null,"Standard error:",0.24786310497140931,0.12031153166154708,9.6164912252682019E-2,0.10752168925089811,0.10646975765777798,4.9676095032696543E-2,5.6653366479370658E-2,6.6268462707466433E-2]
[null,null,null,null,null,null,null,null,null,"Include:",null,null,"Yes","Yes","Yes","Yes","Yes","Yes","Yes","Yes","Yes","Yes","Yes","Yes","Yes","Yes","Yes","Yes","Yes","Yes","Yes","Yes"]
[null,null,null,null,null,"Standardized","Log-",null,"Predicted","Variable:","CMF","Std Err","p1","p2","p3","p4","c1","c2","c3"]
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