Appendix 1 Standard analysis CrossMark 2.0.0 User Manual
Contents
1. var var G 2cov Z 4 can be applied with var 3 var G 184 Table 2 Ancillary intercept predictors Intercept Intercept_obs t Intercept_ pre t 1 2p 3 1 2 3 t 1 1 0 0 0 0 0 1 1 0 0 0 0 0 t 2 1 0 1 0 1 0 0 1 0 1 0 1 0 0 t 3 1 0 0 I 1 0 1 0 0 1 1 1 0 and cov 3 8 representing the estimated variances of b and 6 and their estimated covariance respectively These variances and covariance are given by CrossMark by checking the option Show covariances of parameters in the Estimation Menu If the test outcome leads to not rejecting the null hypothesis the ancillary variables for the predictor in question are no longer needed and the original predictor Age t in the example can be used possibly along with ancillary variables of other predictors for which the hypothesis does not hold The equations above did not include an intercept for simplicity Of course in most applications an intercept will be present and we will have to decide which type of intercept vector s to employ If we have no nonbackcastable predictors the intercept is simply a single vector containing the value 1 for all cases of all cross sections If however nonbackcastable predictors are utilized we may want to estimate one intercept for time observed and another one for preceding time just as was done for Age t in equations 1 and 2 In that case we would have to construct two ancillary time varying in2. Age t Age_obs t Age_pre t 1 2 3 1 2 3 1 2 3 t 1 19 0 0 19 0 0 0 0 0 45 0 0 45 0 0 0 0 0 t 2 37 38 0 0 38 0 37 0 0 21 22 0 0 22 0 21 0 0 t 3 42 43 77 0 O 44 42 43 0 66 67 68 0 0 68 66 67 0 Inc Inc_obs t 1 2 3 t 1 1500 1500 0 0 7300 7300 0 0 t 2 3500 0 3500 0 9400 0 9400 0 t 3 1200 0 0 1200 2200 0 0 2200 observed logit j1 8 Age_ obs t B Age_pre t b Inc_obs t B Age t 0 be Ine p Age t s Ine 3a preceding logit 1 8 Age_ obs t b Age_pre t B Inc_obs t B 0 b Age t 6 0 ps Age t 3b Thus equations 3a and 3b appear to be equivalent to 1 and 2 respectively Since CrossMark uses a single equation for u we employ the generic equation 3 Parameter 6 can be interpreted as 6 i e the effect of age controlled for income at observation time 8 is interpreted like 0 as the effect of age at preceding points in time not controlled for income B has the same interpretation as 6 i e the effect of income controlled for age at the time of observation 183 Instead of 3 way may also use another generic equation in CrossMark logit u 6 Age t Age_obs t 3 Inc_obs t 4 Working out 4 for observation time and preceding timepoints results in observed logit j1 6 Age t Age_obs t 3 Inc_obs t f Age t Age t bs Inc b bs Age t b Inc 4a preceding logit u 0 Age
3. file with fixed mu values for this respondent is the first of the two following lines 316 0 0999 925 0 0 009 Value 316 in the first line refers to the sequence number of the respondent the two 0 values that follow are assigned to u and 4 and the three 9 values indicate that u 4 and u are not fixed but have to be estimated The second line refers to another respondent with sequence number 925 in the data file who was 18 years old at t 5 In this example a file with fixed lambda values need not be specified since only values of u are fixed The file with fixed lambda values must contain one line for each case to which fixed A values are assigned Each line starts with the sequence 189 number of the case in the data file and is followed by as many values 0 1 or 9 as there are cross sections minus 1 since these values relate to A through z T being the total number of cross sections The third example given above concerned the analysis of five wave panel data without inflow and outflow If we assume there are 500 respondents then the data file consists of 2500 lines 500 lines for each wave Suppose a particular respondent has the Y pattern 01100 for 1 5 If the sequence number of the respondent in the first wave is 29 then the other four sequence numbers are 529 1029 1529 and 2029 In the file with fixed mu values and the File with fixed lambda values we have to enter the lines given in the box
4. and 6 are the parameter estimates at the iterations k and k 1 I 7 is the inverse of the Fisher information matrix evaluated at 6 6 and 6LL 66 are the derivatives of the log likelihood with respect to the parameters evaluated at p 6 By default the value of the step size is 0 5 If the log likelihood function has a single mode the optimal value for the step size would be 1 It is not unusual however for the log likelihood function to have multiple modes in which case a step size of 1 could easily cause the algorithm to 178 jump over the parameter region with the highest mode For this reason a default step size of 0 5 is chosen A much smaller step size value may slow down the algorithm too much There is no rule of thumb given here as to the choice of the most efficient step size value The Step size shrinkage s also deals with the problem of the step size being too large If the log likelihood based on 18 lower than the one based on the current step size has apparently been too large In that case CrossMark produces the message Not converging back to parameter estimates of previous iteration and takes as the new step size the product s If this smaller step size also leads to 6 4 estimates with a lower log likelihood than the one based on the step size s s is tried In short the step size is multiplied by s as many times as needed to produce an increase in log likelihood The iterative
5. p _ 1 thus pP y 1 The third and final point concerns the fact that in models for panel data the likelihood is commonly computed for the data of gt 2 while in 188 CrossMark the likelihood for t 1 is used as well To delete the likelihood contribution of the cases for t 1 in CrossMark we assign a very small frequency to the cases of the first wave 1 e 0 0000000001 in the t y x fre data file We can also delete all cases of the first wave from the data file except one case and assign the small frequency value to this single case This single remaining case for t 1 may have any values on the Y and X variables since it only acts as a dummy case having virtually no influence on the parameter estimates 3 1 Specifying fixed u and A values in CrossMark The fields File with fixed mu values and File with fixed lambda values in the Main Menu can be used to enter the names of the data files containing fixed u and A values for some or all cases of some or all cross sections The file with fixed mu values must contain one line for each case to which fixed u values are assigned Each line starts with the sequence number the case has in the t y x fre data file and is followed by as many values 0 1 or 9 as there are cross sections In the first example given above where the age of a respondent say the 316th respondent in the data file was 18 years at the time point of the third cross section the line to enter in the
6. a case of cross section 2 or A for all cases are assigned the missing value 9 By default in CrossMark s output no co variances of parameter estimates are shown They will be if the option Show covariances of parameters is checked before running the model The options for Unobserved heterogeneity and Metropolis sampling will be discussed below in separate sections After clicking the OK button of the Estimation Menu the Main Menu reappears To save all the specifications entered click the Save button and specify a file name e g vote crm which then appears in the top line of the Main Menu Using the Save as button enables saving the job under a different name The most recently saved job can be opened by clicking on the button Last job while older jobs may be opened with Other job To start the analysis the data have to be read first This is done by clicking on Read data When finished reading CrossMark presents the total number of cases as well as the number of cases for each cross section in the rightmost window of the Main Menu After reading the data the estimation can be carried out by clicking on Go The initial log likelihood based on the starting values of the parameters appears on the screen after a few moments as does the log likelihood of each subsequent iteration When the last iteration is finished a Ready message is delivered The estimation may take some time especially when many cases and or predictor var
7. below File with fixed mu values File with fixed lambda values Wave seqnr 4 H H3 H Hs seqnr A A A A nAB WN As can be seen for the data of wave t we specify a fixed ju value in the file with fixed mu values equal to value of Y e g for wave 3 we specify 4 Y 1 The fixed A _ value that has to be specified in the File with fixed lambda values for the data of wave t is equal to the complement of Y 4 Unobserved heterogeneity CrossMark offers the possibility to account for the influence of unobserved variables on the entry and exit probabilities In doing so the assumption is made that the overall contribution of these variables to the logits of the transition probabilities is constant for the time period considered The logit equations for u and 1 A including the contributions of unobserved variabels can be written as follows 190 logit u x6 6 logit 1 28 6 where x is a row vector with the values of the observed potentially backcasted predictors and 6B are the column vectors with the parameters associated with x and finally 6 and represent the total contribution of the unobserved variables The values of 6 and 6 for all respondents or cases are considered to be drawn from a normal distribution with zero mean and variances y en y The above equations therefore can also be written as logit H 28 42 logit 1 26 y2 with z N 0 1 being the standard
8. each respondent s previous state was Y 0 one may truly consider p an entry probability This would e g be the case if political party A did not exist before 1996 In many applications of course the Y 1 state does exist prior to t 1 and 175 respondents could have been in that state In such situations one may prefer to model p as a state probability rather than an entry probability This is accomplished by estimating different sets of parameters for u and for u and following as is done in the model above where the parameters 6 and 6 only apply to 4 In CrossMark the model equations can be specified in the Design mu and Design lambda fields of the Main Menu In Design mu we indicate which predictor variable acts upon which entry probability u For the example this is done as follows uk WNP ooo0oorH oooorH PRRRO PRRRO The first column is the time index and the other four columns correspond to the four predictor variables in the model The second column corresponds to intercept 1 and the value 1 for 1 indicates that intercept 1 has an effect on y the O scores in the second column for t 2 3 4 and 5 indicate that intercept 1 does not have an effect on p l pand u The rightmost column is related to the time varying predictor age the O value for 1 indicates that age does not occur in the equation for u while the 1 values for t 2 3 4 and 5 indicate that age does occur i
9. either 1 or 0 There is no need to aggregate over t X and Y However ageregating the data as is done in this example can speed up the estimation process considerably We now return to the predictor variables X The predictors numbered 1 2 and 3 above are constant over time while predictor 4 takes a different value in each of the five years Time constant predictors occupy a single column in the data file while time varying predictors occupy as many columns as there are cross sections i e five in the example The names and types constant or varying of the predictors have to be specified in the submenu Predictor names and types which shows up after clicking the X names button of the Main Menu and is shown in Figure 2 The left field of the submenu Predictor names and types contains the predictor s name and the right field the predictor s type For a time constant predictor enter the character c and for a time varying predictor enter v Having done so click OK to return to the Main Menu To understand why we use two intercepts and two age predictors instead of just one intercept and one age predictor which would be possible too Figure 2 Predictor names and types Crossmark Predictor names and types Enter predictor names in left window Names may be up to 20 characters long In the right window enter behind each name the predictor type c time constant or v time varying Use one line for each predictor OK intercept
10. for each case It is located in column 2 of the data file In the sequel we will refer to it as intercept 1 2 The respondents age in 1996 located in column 3 For the respondents of the cross sections 1997 and following the age in 1996 has been computed by backcasting their age to the year 1996 We shall explain below why we use age in 1996 as a separate predictor which we call age 1996 3 A second intercept in column 4 which is called intercept 2 4 The respondents age in each of the five years located in columns 5 through 9 These five age values together constitute a single predictor variable the values of which change over time We call this predictor age We will return to the characteristics of the 4 predictors and the way they affect the transition probabilities in more detail below The last two columns 10 and 11 of the data file concern the total number of cases and the number of cases in Y category 1 respectively For example the first record of the cross section at t 5 1 e the record 5 1 18 1 18 19 20 21 22 7 5 173 specifies that there are 7 cases in this cross section who were 18 years old in 1996 19 years old in 1997 etc and that 5 of them are in Y category 1 at t 5 while the other 2 are in category 0 If each row in the datafile would contain data for just a single case then the last but one column here column 10 would be 1 for all cases while the last column would be
11. for the sz probabilities in question for respondents younger than 18 have to be entered in CrossMark A second example of adjusting the basic equations for p is the following Suppose all predictor variables we would like to use are constant over time but only for a short time period To be more specific we assume that the predictor values for a case observed at time t also apply to t 1 and t 2 but not further back in time Therefore we let the Markov chain for each 186 case start two time points preceding to the one the case was observed instead of starting at time point t 1 as we would have done had the predictors been perfectly stable This implies that the first state probability estimated for the cases of the cross section at t 5 will be p For the cases of the cross section at t 4 p will be the first estimated state probability and for those of the cross section at t 3 t 2 and t 1 p will be the first estimated state probability This is different from the more general situation where for all cases of all cross sections p is the first estimated state probability Remember that for p we used a logistic equation p 4 with specific O parameters different from the ones of H through u Here we would like the same to hold for p and p as far as the cases of the cross sections at t 4 and t 5 respectively are involved To achieve this we shall again use the equation p to estimate p as the first est
12. results C Crossmark vote txt Sim data Show paims Go Show resis Stop previous step Estimation is started by pushing button Go The estimated parameter values for all simulated Y datasets are written to the file specified after Output file parameter estimates in the following format the sample number the parameter values of all predictors for the entry probability the parameter values of all predictors for the 1 exit probability and finally the value of the loglikelihood The file specified after Output file for results contains the final results for all simulated Y datasets similar to the ones that are generally shown in the Output window As with the metropolis sampler here again one will have to evaluate the estimated parameters with other statistical software The two buttons Show parms and Show results show the corresponding files in Wordpad 195
13. 2 age Cancel 174 we take a closer look at the model equations for p p Pa p and p or in words the probabilities to vote for political party A in each of the five years In general the basic equations CrossMark uses are with five cross sections P H Py p L A p m P P 1 A 0 p My Py Ps A n My Ps PUA 1 Ba Hs In the example the transition probabilities u and A depend on the respondents ages as follows logit H 61 By A gejoo logit u b b A Jeor logit l A bi B A g 997 logit u B b A geo logit 1 A B B A gerorg logit H4 B b A geo logit A 8 8 Agere logit u 83 B A ges logit 1 A 67 B Agesoo Agejo96 refers to the respondent s age in 1996 Agejo97 to the age in 1997 etcetera The symbol A indicates the exit probability A is the probability not to vote for party A in 1998 given a vote for A in 1997 For the complement of A or the probability to stay in state Y 1 the term exit probability is used in the sequel The symbol u indicates the entry probability jz is the probability to vote for A in 1998 given a not vote for A in 1997 Speaking of u p as an entry probability can be problematic Generally spoken p is the probability to be in state Y 1 at t 1 and this need not to be the same as the probability to be in state Y 1 given that the previous state was Y 0 Only if one knows that
14. CrossMark 2 0 0 A ppenaix User Manual The program CrossMark is designed to estimate transition probabilities using data from repeated cross sections Given a dichotomous Y variable CrossMark estimates the effects of predictor variables X on the entry and exit probabilities using a Markov model CrossMark is available for Windows 95 98 2000 an XP The program needs not be installed simply place file CrossMark exe in a directory of your choice and double click on this file in Windows Explorer to start CrossMark The Main Menu then appears on the screen This menu looks like the one in Figure 1 except that all fields are still empty 1 Standard analysis We shall describe how a standard analysis with CrossMark proceeds using a fictitious example on vote intention To highlight all the options of the program we use bold face characters for buttons that must be clicked and fields or menu s that have to be filled in Suppose the data to be analyzed are from 5 cross sections gathered in consecutive years i e from 1996 to 2000 The dependent variable is the intention to vote for political party A code 1 vote for 0 not vote for and the independent variable is the respondent s age ranging from 18 to 70 years The file containing the data is named c crossmark vote dat This filename has to be entered on the Main Menu in the field Data file t x n f1 The data file can be inspected by clicking the Edit button which o
15. The resulting estimates en G can be interpreted as the effects of the predictors z corrected for the average influence of the unobserved variables Using the above equations and estimation procedure has consequences for the standard errors of 6B and B which can be quite different from the ones estimated without taking into account unobserved heterogeneity The values of and are the estimates of the standard errors of 6 and respectively i e of the contributions of the unobserved variables to the logits of the entry and exit transition probabilities 4 1 Testing the hypothesis H 7 7 0 To test this hypothesis we may use a test procedure described by Snijders and Bosker 1999 We first calculate the value of A 2 loglikelihood for the model including y z and yz Then we compute B 2 loglikelihood for the model without y z and y z and obtain the difference D B A Finally we test the difference D to be significant using a x distribution with 2 degrees of freedom but halve the right tail probability associated with the value of D The standard estimation procedure in CrossMark does not take into account the possible influence of unobserved heterogeneity If we wish to perform an analysis as described above including the y z and y z terms in the equations for the transition probabilities we have to go the Estimation Menu and click on the option called Extra Bernoulli variance After running the model w
16. ability only at the time the respondent was observed but not at preceding points in time We will show using a simple example how such variables can be handled in CrossMark Suppose that we have three cross sections and the nonbackcastable predictor we would like to use is named Inc representing the monthly personal income of a respondent at the time of observation Also we have the backcastable predictor age specified as Age t where the t between brackets denotes that there are three age vectors one for each of the three points in time For simplicity we omit the intercept in the equations for ju below For any respondent of the second and subsequent cross sections the following two equations apply to logit u depending on whether relates to the time the respondent is actually observed or to a preceding point in time 181 observed logit u 6 Age t G Inc 1 preceding logit u B Age t 2 In equation 1 we can use Inc as a predictor whereas in equation 2 this is not possible Of course the Age effects 3 and 6 need not necessarily be the same In order to estimate 6 and 6 with CrossMark a single equation for logit u must be specified that applies to all points in time To achieve this we construct three ancillary time varying predictors which we shall call Age_obs t Age_pre t and Inc_obs t to be discussed below The construction of these predictors must precede the analysis with CrossMark and t
17. e will find the estimates 7 en 7 in the Output window 192 5 Metropolis sampling In the Estimation window an MCMC procedure can be performed that uses pure Meropolis sampling To do so check the option Metropolis sampling and specify a filename after Outputfile posterior parameter values for the file that the sampled parameter points are written to We only implemented this option in a very basic sense There is e g no prior distribution that can be specified for the parameters the implicit prior used for all parameters is the uniform distribution After Length of chain specify the number of samples that has to be drawn from the posterior distributions of all parameters Note that no burn in period can be provided and hence the length of the chain must be large enough to also contain the desired burn in period After pushing button OK CrossMark first performs the usual maximum likelihood ML estimation process Once this is finished the metropolis sampler is started Consequently metropolis sampling begins by default at the ML point To start metropolis sampling from any other parameter point specify the parameter values for this point as the starting values to be used and also set the maximum number of iterations to 0 It is possible to let CrossMark for each sampled parameter point calculate the mean values of p u and A over all cases 2 for each timepoint t To this end a filename must be entered after Outputfile posterio
18. edictors in the second model are defined to be zero and automatically added to the list If a predictor that was present in the previous model does not appear in the second the user has to remove the relevant starting values from both lines If for some reason one would like to fix the parameters of one or more predictors to certain predefined values instead of having them estimated by CrossMark one can be proceed as follows In the field named Fixed entry parameters enter a value 0 or for each predictor parameter that has to be estimated enter 0 or not enter 1 Be sure to enter a value O or 1 for all predictors and to use the same order for the predictors as was used in the menu Predictor names and types For predictors that have a value 0 specified CrossMark will estimate a parameter starting from the starting value For predictors that have a value 1 specified CrossMark will not estimate a parameter but substitutes the given starting value as the parameter value to be used for this predictor s effect on the entry probability In CrossMark s output fixed parameters are denoted by the character f and have a Wald Significance and Std error of 1 0 In the same manner one can fix parameters for the 1 exit probability The Step size field in the Estimation Window refers to the step size of the Fisher scoring algorithm employed for iteratively updating the parameter estimates The algorithm iS given by 6 9 e 1 6LL 66 where 6
19. el for discrete panel data reads as P Y LX a Vata is t 2 5 while for cross sections it reads as R Pel a es t 2 5 the difference being the use of y in the case of panel data and p when using cross sectional data As stated earlier CrossMark uses the second equation since it was designed for the analysis of cross sectional data However the program can simply be tricked to analyze panel data as well and thus to apply the first equation To do so we first have to construct the data file in the way CrossMark expects it to be i e according to the t y x fre format Each cross section in this data file corresponds to a particular wave of the panel data The data for the first wave have to be placed at the top of the data file followed by the data for the second wave the third wave and so on The order in which the respondents appear within the data for each wave is irrelevant and need not be the same for each wave Second we need to define p y _ for t 2 5 or to put it simply p y for t 1 4 To do so we use fixed u and fixed A values To make sure that p y we simply let u y resulting in Dp 4 y For p through p we proceed as follows If for a certain case y 0 t 2 4 we let A 1 and yu 0 which results in P Bala tH 2 eH pH ea paS thus P Y 0 as was meant to be the case If on the other hand y 1 we let A 0 and pw 1 so that p p 1 0 1
20. estimation process ends if either the percentage change in log likelihood is less than the Minimal LogLikelihood Change specified which by default is 0 000001 or the Maximum number of iterations has been reached which by default is 1000 Also by default CrossMark only shows the parameter estimates of the final iteration and not those of previous iterations To force CrossMark showing the estimates of each iteration check the Show iteration history option By default CrossMark applies caseweights resulting in the same weighted number of cases for each cross section The sum of all caseweights is equal to the total number of cases in all cross sections To prevent this weighting procedure uncheck the option Weight cross sections equally CrossMark produces an output file the name of which can be specified in the field Outputfile for t mu lambda p fre By default it is labeled tmulapfre and placed in the directory where the crossmark exe resides The output file contains one line for each case in the data file For case 7 this line has the following information from left to right the time index of the cross section case 7 belongs to the predicted values of u to Hir the predicted values of A to A r 179 the predicted values of p to p p the frequency of case 7 equal to the frequency specified in the rightmost column of the data file Predicted u A and p values that do not apply to a particular case e g 4 for
21. he user must add the predictors to the data file and treat them like any normal predictor variable their names and types v have to be entered using the X names button in the Main Menu and also three columns one for each predictor have to be added to the Design mu and Design lambda matrices The predictor Age_obs t has to be constructed such that Age_obs t Age t for cases observed at time point and Age_obs t 0 for all other cases For predictor Age_ pre t it must hold that Age_pre t Age t for cases observed after time point and Age_pre t 0 for all other cases For 6 randomly chosen cases two of each cross section the values of Age t Age_obs t and Age_pre t might be those shown in the upper part of Table 1 Note that put next to one another the three A ge_ obs t vectors form a block diagonal matrix and the Age_pre t vectors a sub block diagonal one For Inc and Inc_obs t the values of the 6 cases might be the ones in the lower part of Table 1 with now the Inc_obs t vectors forming a block diagonal matrix Instead of the two separate equations 1 and 2 we can write a single equation holding for time observed as well as preceding points in time logit s1 B Age_ obs t B Age_pre t bs Inc_obs t 3 Why 1 and 2 are equivalent to 3 becomes clear when equation 3 is worked out for the observed and preceding time points separately 182 Table 1 Ancillary predictors for Age
22. iables are involved In the mean time the user may want to look at intermediate results by clicking the Show Out button or pressing Ctrl Tab on the keyboard The Output window then appears with the parameter estimates of each iteration scrolling over the screen accompanied by the log likelihood and possibly messages concerning corrective actions undertaken by the estimation algorithm Pressing Ctrl Tab again or 180 clicking the cross X in the upper right corner of the screen closes the Output window Back in the Main Menu the estimation process if still running can be stopped by using the Stop button This may be useful if e g the log likelihood does not change substantially anymore Another reason to stop the iterations is that the algorithm does not converge which may happen if the model contains too many 1 e not uniquely identified parameters To leave CrossMark click Exit or the cross X in the upper right corner of the screen 2 Nonbackcastable variables It may be that the respondent s value on a predictor variable at time is known but the values at t 1 t 2 and so on are not Take e g the variable monthly income Given the income of a respondent of cross section t usually little if anything is know about his or her income at earlier points in time To put it another way the variable income cannot be backcasted Such a nonbackcastable variable can be used as a predictor for the entry and exit prob
23. imated state probability for all cases of all cross sections and then i let p have the same value as p for the cases of the cross section at t 4 and ii let p have the same value as p for the cases of the cross section at t 5 By doing so we estimate three first state probabilities p p and p using the logistic equations p 44 Pp 4 and p u At the same time p and p are also estimated by a Markov equation for the cases of the cross sections at t 3 and t 4 respectively To specify the model we exploit fixed u and A values Let us take a look at a case of the cross section at t 5 for which we want to estimate p using the equation p u4 We let A A 0 and pw yw 0 which results in P H P p 1 A L py My p 11 0 0 p 0 4 Ps P A p m1 0 1 m 0 p Py P A Ds H P p ZA ag 1 Py Hs As can be seen the equations for p and p are the usual Markov equations while for p we have p u For cases of cross section at t 4 we proceed in a similar way by fixing A 0 and u 0 which leads to p 4 For the cases of the cross sections at t 3 t 2 and 187 t 1 we automatically have p 4 so for these cases we do not need to fix any u or A The last example of using fixed u and A values concerns the analysis of discrete panel data Consider a situation in which we have at our disposal a five wave panel data set without any inflow or outflow The Markov mod
24. ized contribution of the unobserved variables and y and y the parameters associated with the predictor z Since the z values for all cases are unknown the parameters 3 6 J en cannot be estimated However given a set of parameter values and the value of z it is of course easy to determine the log likelihood contribution 4l of that case Also for a given set of parameter values the expected or marginal log likelihood contribution E 0 of a case can be determined where the expectation is taken over all possible values of z taken from N 0 1 For a case of e g the cross section at t 2 it holds that EUO f pr 0 p m f e dz ify 1 and E Q m fe dz ify 0 00 Here u and A are defined as above i e including z p is defined as usual i e p 4 without z in CrossMark controlling for unobserved variables is only possibly for the transitions probabilities at t gt 2 and f z is the height of the standard normal pdf at z The integrals cannot be derived analytically but are approximated by CrossMark using Gaussian quadrature with 20 mass points Utilizing the F C values of all cases of 191 all cross sections it is possible to estimate those values 6 6 4 en 4 that averaged over all values that z can take have the highest expected or marginal log likelihood The criterion to maximize in this estimation is the sum of the E C values of all cases of all cross sections
25. n the equations for H H3 y and ty In general the Design mu matrix must have as many rows as there are cross sections Each row starts with the time index and is followed by a 1 or 0 value for each predictor variable indicating whether 1 or not 0 the predictor acts upon entry probability p In the same way a Design lambda matrix has to be specified indicating which predictor acts upon which exit probability A For the present example the lambda matrix is specified as Ook WNP ooo Momo ooo0oo0oo0 PRRRO PRRRO 176 Note that the first row of the Design lambda matrix contains the value 1 for the time index t 1 and else only 0 values to indicate that none of the four predictor variables has an effect on A This is just to specify that A does not play a part in the model equations We proceed by clicking the Estimation button of the Main menu to invoke the Estimation Menu as shown in Figure 3 The upper two fields in this Estimation Menu specify the starting values for the iterative Fisher scoring scheme The default values are 0 for all 3 and 8 parameters of the entry and 1 exit probabilities respectively Good starting values i e values close to the final ML estimates speed up the estimation process Starting values far removed from the final estimates slow down this process or may cause the estimates to be caught in a local maximum or not to reach convergence at all When convergence has been reached it is advisable to cho
26. ose other starting values and let CrossMark run again to check whether the same parameter estimates are found If this turns out to be the case one can be more confident that the estimates are indeed the true global ML estimates instead of estimates associated with a local maximum Figure 3 Estimation Menu Crossmark Estimation Menu Starting values entry parameters Starting values 1 exit parameters Fixed entry parameters Fixed 1 exit parameters Step size 0 5 Stepsize shrinkage 9 5 read starting values Minimal LogLikelihood Change 10 000001 Maximum number of iterations 1000 Show iteration history Cancel Weight cross sections equally Outputfile for t mu lambda p fre tmulapfre dat Edit Show covariances of parameters D Unobserved heterogeneity Li Metropolis sampling Length of chain Outputfile posterior parameter values Edit Dutputfile posterior mean p mu lambda Covariance matrix of jumping distribution equals times estimated covariance matrix of parameters 177 When analyzing complex models in the sense of having many predictors starting values become more of an issue The final estimates of a previous relatively simple model can be used as starting values for a new model having additional predictors To this end the button read starting values can be helpful After clicking the final estimates of the previous model are filled in as starting values in both fields The starting values for the additional pr
27. pens the data file in WordPad The total number of cross sections 5 has to be 171 Figure 1 Main Menu Crossmark Main Menu C Crossmark vote crm Data file t x n f1 fc crossmark vote dat Edit Number of cross sections 5 X names File with fixed mu values Edit File with fixed lambda values Edit Design mu Reading data Total number of individual cases 2419 Number of individual cases in each crosssection 484 cases 463 cases 465 cases 578 cases 429 cases Starting iterations now Initial Loglikelihood 910 76385794 Iteration 1 loglikelihood 577 4394334530 Design lambda Iteration 2 loglikelihood 468 7246899134 Iteration 3 loglikelihood 421 6526986909 Iteration 4 loglikelihood 404 5641100540 Iteration 5 loglikelihood 398 5805607658 Iteration 6 loglikelihood 396 6360220116 Iteration 7 loglikelihood 396 0530348884 Iteration 8 loglikelihood 395 8895157295 Iteration 9 loglikelihood 395 8456770100 Iteration 10 loglikelihood 395 83422801 20 Iteration 11 loglikelihood 395 831 2772325 Iteration 12 loglikelihood 395 8305208808 Iteration 13 loglikelihood 395 8303272392 Iteration 14 loglikelihood 395 8302776046 Iteration 15 loglikelihood 395 8302648504 Iteration 16 loglikelihood 395 8302615623 Ready Last job Other job Save Save as Estimation Simulate Read data Go Show ou Stop Exit entered in the field Number of cross sec
28. r mean p mu lambda The value to be entered on the Estimation window in the sentence Covariance matrix of the jumping distribution equals times estimated covariance matrix of parameters refers to what is discussed by Gelman Stern and Rubin in Bayesian data analysis 1995 on page 334 at the bottom where c 2 4 sqrt d Value 2 4 for c is the default CrossMark uses if you don t specify another value in the above sentence After the metropolis sampler is finished inspection of the chain of sampled parameter points in the file specified after Output 193 posterior parameter values is always recommended to make sure that the chain has changed fast enough If the same parameter points are resampled many times a smaller value for c is probably more appropriate The parameter values that are sampled by the metropolis algorithm are written out to the file specified after Outputfile posterior parameter values in the following format sequence number 1 2 3 4 100000 or more depending on the length of the chain that was entered followed by the values of the parameters of all predictors on the entry probabilities followed by those on the 1 exit probabilities followed finally by the loglikelihood value associated with these parameter values To evaluate the output files with posterior parameter values and or means of Pa Ha and Ap other statistical software must be used CrossMark itself does not perform any chain evalua
29. t Age_obs t Inc_obs t logit u 6z Aget 0 6 0 logit su 3 Age t 4b As can be seen 4a is equivalent to 3a and 1 while 4b is equivalent to 3b and 2 Therefore both equation 3 and 4 can be used to model logit 4 They differ only in parameterization The sum 6 3 has the same interpretation as 6 or 6 6 is interpreted in the same way as 6 or 8 Finally the interpretation of is similar to the one of 8 or A minor advantage of using 4 instead of 3 is that 4 needs on construction of the Age_pre t vectors 2 1 Testing the null hypothesis H 3 6 Looking at the equations 1 and 2 the question arises as to the equality of the two Age effects 6 and 6 When applying equation 4 the above null hypothesis translates into H 6 or more simply to H 6s 0 This test is automatically performed by CrossMark and the significance level of the related Wald statistic is reported in the Output window When on the other hand equation 3 is applied the above hypothesis translates into H 3 6 0 Given the hypothesis is true the sample outcome of the statistic G var with var G G being the estimated sample variance of 3 follows a x distribution with 1 degree of freedom The value of p can of course be derived from the ML estimates produced by CrossMark in the final iteration To derive var G the formula var
30. tercept predictors according to the scheme in Table 2 185 3 Fixed u and values CrossMark has the option of entering fixed u and or fixed A values for some or all cases on some or all points in time We start with discussing three situations in which this option can be utilized to adjust the basic equations for the state probabilities p We also explain how the option has to be specified in CrossMark In some applications the values for u and or A may be considered fixed and hence need not be estimated This would e g be the case when the backcasted age of a respondent is 17 or younger in a study on voting behavior given that the voting age is 18 Suppose in the example given earlier a respondent is 18 years old at the time that the third cross section was observed i e on 3 For this respondent we would like p and p to be zero also since p is an entry probability the respondent could not have voted for party A at t 2 we would like p to equal the entry probability u To implement these restrictions in the model equations we fix u m 0 for this respondent which implies the following adjusted equations for p to p P 6 0 Py pL A A p m O1 A 1 0 0 fle i ee ll h p 1 A l ps Ma p 1 A 1 py Hs The equations for p and p have the usual Markov form while those for P P and p are adjusted in the sense specified above We shall explain below how the fixed Qvalues
31. tion produces no histogram s of posterior parameter estimates and or means nor calculates means or standard deviations of the samples that were taken from the posterior distributions 6 Parametric bootstrap The Simulate button on the Main Menu opens the Simulate window where a parametric bootstrapping procedure can be performed This window is shown in Figure 4 In the first step of the parametric bootstrap procedure anumber of Y datasets are simulated based on the observed X values in the data file and a set of true parameter values that must be specified after True values entry parameters and True values 1 exit parameters The number Y datasets that have to be simulated is specified after Number of simulations A name is generated automatically but can be modified for the output file that will contain the simulated Y data After clicking the button Sim data the simulation process starts during which the Y data are generated and written to the file specified Once the simulation has been finished the next step can be started during which the parameters will be estimated for samples that were simulated in the 194 Figure 4 Simulate window Crossmark Simulate Output file simulated Y data C Crossmark vote Y dat Number of simulations 5000 read starting values True values entry parameters 3 01 2 006 True values 1 exit parameters oO 4 003 Output file parameter estimates f Crossmark vote parms dat Output file for
32. tions The abbreviation t x n f1 behind Data file stands for t time index x X or predictor variables n number of cases and fl number of cases in category Y 1 respectively and reflects the order in which the data must appear in the data file The first three lines of the example data of each cross section are shown below 1 1 18 1 18 19 20 21 22 9 2 1 1 19 1 19 20 21 22 23 5 0 1 1 20 1 20 21 22 23 24 3 0 2 1 18 1 18 19 20 21 22 4 1 2 1 19 1 19 20 21 22 23 13 5 2 1 20 1 20 21 22 23 24 8 0 3 1 18 1 18 19 20 21 22 4 2 3 1 19 I L20 208 22 23 5 1 3 1 20 1 20 21 22 23 24 8 3 4 1 18 1 18 19 20 21 22 4 2 172 4 TL9 1 19 20 21 22 23 12 11 4 1 20 1 20 21 22 23 24 8 5 5 1 18 1 18 19 20 21 22 7 5 5 TL9 1 19 20 21 22 23 4 1 5 1 20 1 20 21 22 23 24 2 1 The first data column is the time index As there are five cross sections the time index has to have the values 1 2 3 4 and 5 denoting the years 1996 1997 1998 1999 2000 respectively CrossMark expects the data to be ordered in time with the data of the first cross section located at the top of the file those of the second cross section following underneath and so on until the data of the last cross section which must be located at the end of the file The next 8 data columns of the data file in this example 1 e column 2 through 9 contain the values of the predictor variables X There are 4 predictor variables here n An intercept having the value 1
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