Introduction

Transcription

1 =============================================================================== mla.doc MLA /31/97 =============================================================================== Copyright (c) by Frank Busing Introduction Thank you for your interest in MLA, software for multilevel analysis of data with two levels. The main goal of MLA is to provide a program with an easy-to-use interface, alternative estimation methods and extensive resampling options. This file contains information about the following topics: - running MLA - syntax - comments - statements - summary Running MLA MLA runs as a stand-alone batch program. It uses an input file and an output file as parameters. This means that the program can be started by the command MLA [-hhv] <inputfile> <outputfile> where <inputfile> should be replaced by the name of the input file and <outputfile> replaced by the name of the output file. The options are help (-h), extended help (-H) and verbosity (-v), respectively. Both input- and output files are simple text files (ASCII). Syntax The input file consists of statements, which are case INsensitive. Every statement begins with a slash and a keyword (e.g., /TITLE). Every keyword may be abbreviated, but it must be at least of length three to be recognized. Other text following the keyword and/or leading spaces will be ignored. The rest of the statements must follow on lines below the keyword and should precede the next statement. These lines are called substatements and may also consist of one or more keywords (e.g., FILE). The last statement to be read is the /END statement. All other statements, and corresponding substatements, may appear in any order (but before the /END statement if they are to be reckoned with). A substatement may continue on the next line. In this case the first line must be ended with two backslashes (\\). Comments Comments are preceded by a percent sign (%) and may appear throughout the input file. All text on a line, after and including the percent sign, will serve as comment and is ignored as program input. Statements

2 /TITle (optional) Following the keyword /TITle, the first non-blank line contains the title for the analysis. The title is repeated on top of every part of the output. /DATa (required) The /DATa statement contains information about the data file. This statement has six substatements, three of which are required. FILe (required) This substatement indicates the name of the data file. The name is given after the equals sign and must satisfy the usual DOS conventions on filenames. The file itself is a free-field formatted numbers-only ASCII file. This means that values of variables must be separated by at least one blank. The file must consist of one case per line. Cases must be sorted by the level-2 identifier variable (see below). VARiables (required) The VARiables substatement specifies the number of variables in the data file. ID1 (optional) One of the variables in the data file MAY contain a code (number) that identifies the level-1 units. The number is used in the output to label level-1 units. The variable number has to follow the keyword ID1 and it must indicate the position of the identifier variable in the data file. The variable number must be at least 1 and less than or equal to the number of variables, indicated on the VARiables substatement. If omitted, the order in which the level-1 units are read from the data file is used as label. ID2 (required) One of the variables in the data file MUST contain a code (number) that identifies the level-2 units. The number is essential for a correct discrimination of the level-2 units. Cases must be sorted by the level-2 identifier variable identified on this substatement. MISsing (optional) For every variable, one missing value may be specified on this substatement. After the equal sign, first the variable is indicated followed by the missing value between parenthesis. More variables and values are seperated by comma's. CENTER (optional) CENTER means Center Grand Mean (Kreft and de Leeuw, 1996). Following the CENTER substatement, the variables must be specified, which will be centered (ingnoring grouping) just after reading the data, but before any analysis. More variables are seperated by comma's. /MODel (required) The /MODel statement is followed by a set of equations that specify the model that has to be estimated. There is only one level-1 equation, but there may be one or more level-2 equations. The order in which the level-1 and level-2

3 equations appear is arbitrary. The terms used in the level-1 equation are: - V = variable, which is a variable in the data file. V may be either indicating the outcome variable (V in front of the equal sign) or a predictor variable (V following the equal sign). - B = beta component. At level-1 these are the regression coefficients that seem to be outcome variables at the second level. - E = the level-1 random term. This term is considered to be a residual or error term. The variance of this term has to be estimated from the data. The level-2 equations partly consist of the same terms, but also of specific level-2 equation terms: - B = beta component, corresponding with the level-1 regression coefficient. At this level, however, B can be viewed as an outcome variable. - G = gamma component. These are the fixed parameters to be estimated in the multilevel model. - V = one of the variables from the data file (as explained above). - U = level-2 random term. As with the first level, this component is considered a residual or error term, but now for the second level. The second level may have more than one error term: one for each level-2 equation (i.e., for each beta element). The variances and the covariances of these terms have to be estimated from the data. In the equations each term is directly followed by a number (except for the level-1 random term E). For the V term this number is the variable number, the position of the variable in the data file (e.g., V4, the fourth variable in the data file). The other terms only use a number for identification, without any additional meaning (e.g., G3, one of the fixed parameters). The B terms have meaning in the equations of both levels. Every equation consists of one term before and at least one term after the equals sign. Terms on the right hand side of the equations are connected by plus signs. A variable and a corresponding parameter are connected by an asterisk (*). This is used to connect a fixed parameter and a predictor variable in level-2 equations and to connect a level-1 regression coefficient and a predictor variable in the level-1 equation. /CONstraints (optional) The /CONstraints statement allows fixed paramters to be fixed to a certain value. Constraints can thus only be applied to fixed parameters and then of the form: fixed paramter equals certain value, for example G2=2.0. The fixed paramter is held fixed during estimation and is not used for estimation of the standard errors either. The standard error will be zero and no t-test is performed for this parameter. /TEChnical (optional) The /TEChnical statement provides useful possibilities to alter the estimation process. If this statement and subsequent substatements are not specified, the program will run using default values. ESTimation method (optional) The substatement ESTimation method provides the opportunity to set the estimation method. One can choose between FIMl and REMl. FIMl is the default method and represents full information maximum likelihood estimation. REMl is restricted maximum likelihood estimation.

4 MINimization method (optional) This substatement sets the minimization method. One can choose between BFGS, using the Broyden-Fletcher-Goldfarb-Shanno variant of Davidon-Fletcher-Powell quasi-newton minimization method, and EM, the Expectation-Maximization method. The default minimization method is BFGS. REParameterization (optional) The level-2 covariance matrix should be a positive (semi-)definite matrix. To impose this restrictions, the parameters can be written in the following way: C=LDL', where C is the covariance matrix, L is a lower triangular matrix and D a diagonal matrix. The elements of the diagonal matrix D may either be square ROOTs or powers of e (the complementary of the natural LOGarithm). On default ROOT reparameterization is performed. WARnings (optional) If the maximum number of warnings is reached, the program terminates execution. This substatement can change the default value of 25. The value must be an integer between 1 and MAXimum number of iterations (optional) The default value of MAXiter is 100. This number should be sufficient for reaching convergence if the sample size is large enough and/or the number of parameters to be estimated is not too large. Changing the minimization method or the convergence criterion (see below) can make it necessary to raise the maximum number of iterations. The value must be an integer between 1 and CONvergence (optional) After each iteration the new function value is compared to the previous function value. The obtained difference is compared to a convergence related value. If F[i-1]-F[i] /{0.5*( F[i] + F[i-1] )} <= convergence, convergence is said to have been reached. In this formula, F[i] is the function value after the i-th iteration. The default value of CONvergence is 1.0E-08 and permitted values range from 0.0 to 1.0. SEEd (optional) For diagnostic purposes, one can provide an initial number (seed) for the random number generator. This is specified by the substatement SEEd. Using the same initial seed, the simulation samples will be identical. The seed value must be an integer between 1 and 1,073,735,823. LUXury (optional) Uniform deviates are obtained with te RANLUX pseudo-random number generator (Luscher, 1994). For this generator several types of LUXury may be specified. Five standard levels are defined: 0 = very long period, but fails many tests, 1 = considerable improvement, but still fails some tests, 2 = passes all tests, but theoretically still defective, 3 = default 4 = highest luxury, all 24 bits of mantissa thoroughly chaotic. A higher luxury level also means a slower generation of uniform deviates. FILe (optional) Results of the current analysis are written to file. The file must be a valid filename and be specified after the file keyword. Only the essential

5 information is written to this file. It's content changes over time and some self-inspection will show what is written where. /SIMulation (optional) Several options for simulation are available in MLA. These include jackknife, bootstrap and permutation. Theoretical details concerning the implementation of these resampling methods for the two-level model can be found in the MLA manual (Busing, Meijer, and van der Leeden, 1994). KINd (required) With this substatement the user can choose from three options, namely BOOtstrap, JACkknife and PERmutation simulation. All types of simulation work as follows: 1. perform analysis 2. obtain a (new) sample 3. repeat the analysis 4. save the (new) estimates The last three steps, together called a replication, are repeated a number of times. Afterwards, bias-corrected estimates of model parameters and nonparametric estimates of standard errors are computed. These estimates are computed from the set of saved estimates and the original maximum likelihood estimates. METhod This substatement specifies the method of bootstrap to be performed. It is required whenever KINd = BOOtstrap. One can choose between three different methods: 1. RESiduals (or ERRor). This method resamples the elements of the level-1 and level-2 residuals. Subsequently a new outcome or dependent variable is computed using these residuals, the original predictor or independent variables and the parameter estimates (fixed components). 2. CASes. Using this method a bootstrap sample is created by resampling the original data. Thus, complete cases are randomly drawn (with replacement) from the original cases. The procedure follows the nested structure in the data, by a nested resampling of cases: level-2 units are randomly drawn (with replacement) and cases within a particular drawn level-2 unit are drawn (with replacement). 3. PARametric. This method computes a new outcome or dependent variable using the original predictor variables, the parameter estimates and a set of level-1 and level-2 residuals. The residuals are drawn from a normal distribution with mean zero and variance sigma squared for the level-1 residuals, and from a (multivariate) normal distribution with zero mean vector and covariance matrix theta for the level-2 residuals. TYPe The substatement type is only required whenever the substatement KINd = BOOtstrap is used in combination with METhod = RESiduals. The TYPe substatement specifies the type of estimation that is used to determine the level-1 and level-2 residuals. One can choose between RAW and SHRunken. BALancing (optional) For the bootstrap methods RESiduals and CASes, a balanced bootstrap can be specified on this substatement. In that case BALancing = BALanced must be specified. Default is BALancing = UNBalanced.

6 RESample (optional) The substatement RESample offers the user the choice at which level units will be resampled. The default is 0, which means that at both levels units will be resampled. If KINd = JACkknife, or KINd = BOOtstrap and METhod = CASes, the user may choose 1 or 2, which means that only level-1 units or only level-2 units will be resampled, respectively. LINking The level-1 and level-2 residuals can be drawn linked or unlinked during simulation. Linking the residuals means that the level-1 residuals will be drawn from the same unit as where the level-2 residual was drawn from. This is specified with LINking = LINked. Specifying LINking = UNLinked has the same result as not using the substatement at all. This is the default. REPlications (optional) Using the substatement REPlications the number of bootstrap replications is specified. It must be an integer value between 1 and The default value is 100. CONvergence (optional) See the /TEChnical statement. Specifying the CONvergence substatement within the /SIMulation statement has only implications for the convergence during simulation. FILe (optional) Results of the simulation analysis can be written to a file. Using the substatement file, a filename may be specified. Filenames must satisfy the usual DOS conventions on filenames. /INTerval (optional) Several options for confidence interval estimation are available in MLA. These include normal interval, percentile, bias corrected percentile and bootstrap-t. KINd (required) With this substatement the user can choose from four methods, namely NORmal, PERcentile, BIAs-corrected percentile and BOOtstrap-t. WEIght (optional) This substatement has implications for the internal bootstrap, performed on the bootstrap-t confidence interval estimation. A balanced bootstrap can be specified on this substatement. In that case WEIght = BALanced must be specified. Default is WEIght = UNBalanced. REPlications (optional) As for the previous substatement, this substatement has also implications for the bootstrap-t method. The number of internal bootstrap replications is specified. It must be an integer value between 1 and The default value is 25. ALPha (optional) ALPha is the confidence level (two-sided). Now, the confidence interval is

7 equal to 100(1-2*Alpha). The default value of ALPha is CONvergence (optional) See the /TEChnical statement. Specifying the CONvergence substatement within the /INTerval statement has only implications for the convergence during interval estimation. FILe (optional) Results of the interval estimation can be written to a file. Using the substatement file, a filename may be specified. Filenames must satisfy the usual DOS conventions on filenames. /PRInt (optional) The /PRInt statement gives the user control over the output. Not all output is optional. The default output consists of a title page, an echo of the input, and system information. Output for the simulation analysis is generated whenever the /SIMulation statement is used. INPut If INPut = YES then the input information is digested and displayed in two parts. A required and an optional part. Default is NO. DEScriptives After the keyword descriptives the user may specify both variables and level-2 identification codes. For the total sample size and for every level-2 unit specified the following statistics are computed and displayed: mean, standard deviation, variance, skewness, kurtosis, Kolmogorov- Smirnov's Z, significance level of K-S's Z, minimum, 5th-quantile, first quartile, median, third quartile, 95th quantile and the maximum. RANdom level-1 coefficients The random level-1 coefficients or level-2 outcomes consist of ordinary least squares estimates per level-2 unit. After the keyword B's and sigma may be specified. OLSquares This part contains the ordinary least squares estimates for the fixed (gamma's) and random (variances and covariances of U and E) parameters. A regression analysis is performed, ignoring grouping. For the level-1 error variance two estimates are displayed, the one-step (E(1)) and two-step (E(2)) estimate. RESiduals After the keyword both level-1 and level-2 residuals may be specified (U and E). For the first level, three different types of residuals are displayed, namely the total, raw, and shrunken residuals. The level-2 residuals are the raw and shrunken residuals for the specified level-2 components. These estimates are based on the BFGS-FIML estimates. POSterior means Displayed are the posterior means which are specified following the keyword. These estimates are based on the BFGS-FIML estimates.

8 DIAgnostics For diagnostic purposes the Mahalanobis distances for the level-2 residuals are displayed. /PLOt (optional) The /PLOt statement gives the user control over some plot options. HIStograms This options is only in effect when \SIMULATION is chosen. If so, all parameters may be specified and histograms will be displayed of the specified parameters. SCAtters Scatterplots can be obtained for prediction and residuals. Specifying prediction produces a scatterplot of the response variable versus the predicted values based on the estimated fixed parameters. Specifying a variable produces a scatterplot of this variables versus all residuals associated with this variable. Summary Explanation of the codes used below. <a> means alpha numeric. <filename> means a filename has to be specified. <i> means an integer, possibly followed by the default value. <f> means a floating point, possibly followed by the default value. [a B] means choose between a and b with B as default.,... means that more of the same may occur. /TITLE <a> /DATA file variables id1 id2 missing center /MODEL V<i> = E V<i> = B<i> + E V<i> = B<i> + B<i>*V<i> E = <filename> = <i> = <i> = <i> = V<i>(<f>),... = V<i>,... (one of these equations) B<i> = G<i> B<i> = G<i> + G<i>*V<i> +... B<i> = U<i> ((n)one or more of these equations) B<i> = G<i> + U<i> B<i> = G<i> + G<i>*V<i> U<i>... /CONSTRAINTS G<i> = <f>... /TECHNICAL estimation method = [FIML reml] minimization method = [BFGS em]

9 reparameterization = [none ROOT logarithm] warnings = <i25> maximum number of iterations = <i100> convergence = <f > seed = <i> luxury = <i3> /SIMULATION kind = [bootstrap jackknife permutation] method = [residuals cases parametric] type = [raw shrunken] resample = [0 1 2] balancing = [UNBALANCED balanced] linking = [UNLINKED linked] replications = <i100> convergence = <f > file = <filename> /INTERVAL kind = [normal percentile bias-corr. bootstrap-t] replications = <i25> alpha = <f0.10> convergence = <f > file = <filename> /PRINT input = [yes NO] descriptives = V<i>,...,<i>,... random level-1 coefficients = B<i>,...,sigma olsquares = [yes NO] residuals = U<i>,...,E posterior means = B<i>,... diagnostics = [yes NO] /PLOT histograms = G<i>,...,U<i>*U<i>,...,E scatters = prediction,v<i>,... /END An annotated example can be found in MLA.IN. --- end of mla.doc

10 Dr. Wolfgang Langer - Methoden V: Grundlagen der Mehrebenenanalyse - WiSe 2000/ MLA 3.2 Syntaxchart: (MLA.IN) /TITLE % optional title MLA version 3.2: annotated example /DATA % specification of data file = mla.dat % data file vars = 6 % total number of variables in data file id1 = 3 % Level-1 identification code variable number id2 = 2 % Level-2 identification code variable number missing = v4( ) % missing value variable 4 = center = v6 % center grand mean level-1 predictor variable 6 /MODEL % model specification b1 = g1 \\ % lines may be broken + g2*v6 \\ % using two backslashes + u1 % intercept: level-2 equation 1 b2 = g3 + g4*v6 + u2 % slope: level-2 equation 2 v4 = b1 + b2*v5 + e % level-1 equation /TECHNICAL estimation = fiml % additional, technical specifications % full information maximum likelihood estimation minimization = bfgs % minimization method is dfp:bfgs reparam = root % reparameterization c=ll' warnings = 50 % maximum warnings raised to 50 maximum = 500 % raise maximum number of iterations seed = % initial seed to be used luxury = 4 % increase luxury level for random number generator convergence = 1.0E-12 % set convergence criterion /SIMULATION % specify resample simulation kind = bootstrap % bootstrap simulation analysis method = residuals % resample residuals type = shrunken % use shrunken residuals balance = unbalanced % no balance in resampling linking = unlinked % no linking of level-2 and level-1 residuals replications= 200 % set #replications to 200 /INTERVAL % interval estimation kind = bias-corrected % bias-corrected percentile interval alpha = 0.05 % interval width of 0.90 /PRINT % additional print specification inp = yes % print input specifications des = v4,v5,v6,2,3 % print descriptives of v4,v5,v6, level-2 units 2,3 ols = yes % print ordinary least squares estimates ran = all % print all random level-1 coefficients res = u1,u2 % print residuals u1 and u2 pos = all % print all available posterior means dia = yes % print diagnostics /PLOT hist = g2,g3,g4 scat = pred,v6,v5 /END % additional plot specification % plot histograms of g2, g3 and g4 % plot scatterplots: prediction- and residual-plots % final statement: the end.

11 Dr. Wolfgang Langer - Methoden V: Grundlagen der Mehrebenenanalyse - WiSe 2000/ Beispieldaten für MLA.IN (MLA.DAT)

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15 Dr. Wolfgang Langer - Methoden V: Grundlagen der Mehrebenenanalyse - WiSe 2000/ Ausgabeprotokoll von MLA32 (MLA.OUT): MMMM MMMMM LLLL MMMMM MMMMMM LLLL AA MMMM M MMMMMMM LLLL MMMM MM MMM MMMM LLLL MMMM MMMM MMMM LLLL MMMM MM MMMM LLLL AA MMMM M MMMM LLLL MMMM MMMM LLLL MMMM MMMM LLLL MMMM MMMM LLLL MMMM MMMM LLLLLLLLLLLLLLLLLLLLLLLLLLLL MMMM MMMM LLLLLLLLLLLLLLLLLLLLLLLLLLLLLL Multilevel Analysis for Two Level Data Version 3.2 Developed by Frank Busing Erik Meijer Rien van der Leeden Published by Leiden University Faculty of Social and Behavioural Sciences Department of Psychometrics and Research Methodology Wassenaarseweg 52 P.O. Box RB Leiden The Netherlands Phone +31 (0) Fax +31 (0)

16 Dr. Wolfgang Langer - Methoden V: Grundlagen der Mehrebenenanalyse - WiSe 2000/ MLA (R) Multilevel Analysis for Two Level Data Version Copyright Leiden University All Rights Reserved Part 1 Wed Oct 11 14:07: Inputfile statements 1 /TITLE % optional title 2 MLA version 3.2: annotated example 3 /DATA % specification of data 4 file = mla.dat % data file 5 vars = 6 % total number of variables in data file 6 id1 = 3 % Level-1 identification code variable number 7 id2 = 2 % Level-2 identification code variable number 8 missing = v4( ) % missing value variable 4 = center = v6 % center grand mean level-1 predictor variable 6 10 /MODEL % model specification 11 b1 = g1 \\ % lines may be broken 12 + g2*v6 \\ % using two backslashes 13 + u1 % intercept: level-2 equation 1 14 b2 = g3 + g4*v6 + u2 % slope: level-2 equation 2 15 v4 = b1 + b2*v5 + e % level-1 equation 16 /TECHNICAL % additional, technical specifications 17 estimation = fiml % full information maximum likelihood estimation 18 minimization = bfgs % minimization method is dfp:bfgs 19 reparam = root % reparameterization c=ll' 20 warnings = 50 % maximum warnings raised to maximum = 500 % raise maximum number of iterations 22 seed = % initial seed to be used 23 luxury = 4 % increase luxury level for random number generator 24 convergence = 1.0E-12 % set convergence criterion 25 /SIMULATION % specify resample simulation 26 kind = bootstrap % bootstrap simulation analysis 27 method = residuals % resample residuals 28 type = shrunken % use shrunken residuals 29 balance = unbalanced % no balance in resampling 30 linking = unlinked % no linking of level-2 and level-1 residuals 31 replications= 200 % set #replications to /INTERVAL % interval estimation 33 kind = bias-corrected % bias-corrected percentile interval 34 alpha = 0.05 % interval width of /PRINT % additional print specification 36 inp = yes % print input specifications 37 des = v4,v5,v6,2,3 % print descriptives of v4,v5,v6, level-2 units 2,3 38 ols = yes % print ordinary least squares estimates 39 ran = all % print all random level-1 coefficients 40 res = u1,u2 % print residuals u1 and u2 41 pos = all % print all available posterior means 42 dia = yes % print diagnostics 43 /PLOT % additional plot specification 44 hist = g2,g3,g4 % plot histograms of g2, g3 and g4 45 scat = pred,v6,v5 % plot scatterplots: prediction- and residual-plots 46 /END % final statement: the end. 46 lines read from "mla.in"

17 Dr. Wolfgang Langer - Methoden V: Grundlagen der Mehrebenenanalyse - WiSe 2000/ MLA (R) Multilevel Analysis for Two Level Data Version Copyright Leiden University All Rights Reserved Part 2 Wed Oct 11 14:07: MLA version 3.2: annotated example Input information Required Name of datafile : MLA.DAT Number of variables : 6 Level-2 id. column : 2 Equation 1 : B1=G1+G2*V6+U1 Equation 2 : B2=G3+G4*V6+U2 Equation 3 : V4=B1+B2*V5+E Single equation : V4=E0+G1+G2*V6+G3*V5+G4*V6*V5+U1+U2*V5 Optional Title of analysis : MLA version 3.2: annotated example Level-1 id. column : 3 Missing for var 4 : Center variables : V6 Estimation method : 1 Minimization method : 1 Reparameterization : 1 Maximum iterations : 500 Convergence : 1e-12 Warnings (maximum) : 50 Kind of simulation : 1 Simulation method : 3 Simulation balance : 0 Simulation linking : 0 Residuals type : 2 Resampling type : 0 Luxury level : 4 Initial random seed : Simulation convergence : 1e-10 Number of replications : 200 Simulation output file : Kind of CI estimation : CI alpha : CI convergence : 1e-10 CI replications : 25 Print input : 1 Print explore : 1, V4,V5,V6,2,3 Print olsq : 1 Print outcomes : ALL Compute residuals : 1 Print residuals : U1,U2 Print posterior means : ALL Print diagnostics : 1 Print intervals : 1 Max equations : 3 Level-1 size : 2 Level-2 : 2 X-size : 4 Z-size : 2 Parameters : 8 Level-2 parameters : 3 Input file : mla.in Output file : mla.out Verbose : 0 Monte Carlo : 0

18 Dr. Wolfgang Langer - Methoden V: Grundlagen der Mehrebenenanalyse - WiSe 2000/ Monte Carlo file : Plot histograms : G2,G3,G4 Plot scatterplots : PRED,V6,V5 Response variable : 4 Explanatory variables : 0(1) 6(2) 5(3) 6(4) Random level-2 vars. : 0(1) 5(2) Random level-1 coeffs. : 0(1) 5(2) Level-2 outcome 1 : 0(1)[1] 6(2)[1] Level-2 outcome 2 : 0(3)[2] 6(4)[2] MLA (R) Multilevel Analysis for Two Level Data Version Copyright Leiden University All Rights Reserved Part 3 Wed Oct 11 14:07: MLA version 3.2: annotated example Data descriptives Data descriptives for all units # Level-1 units = 231 # missing Level-1 units = 1 # correct Level-1 units = 230 # correct Level-2 units = 15 Var Mean Stddev Variance Skewness Kurtosis K-S Z Prob(Z) Var Minimum P5 Q1 Median Q3 P95 Maximum Data descriptives for level-2 unit 2 # Level-1 units = 16 Var Mean Stddev Variance Skewness Kurtosis K-S Z Prob(Z) Var Minimum P5 Q1 Median Q3 P95 Maximum Data descriptives for level-2 unit 3 # Level-1 units = 18 Var Mean Stddev Variance Skewness Kurtosis K-S Z Prob(Z)

19 Dr. Wolfgang Langer - Methoden V: Grundlagen der Mehrebenenanalyse - WiSe 2000/ Var Minimum P5 Q1 Median Q3 P95 Maximum MLA (R) Multilevel Analysis for Two Level Data Version Copyright Leiden University All Rights Reserved Part 4 Wed Oct 11 14:07: MLA version 3.2: annotated example Random Level-1 coefficients: ordinary least squares estimates per level-2 unit Parameter B1 Unit Size Estimate SE T Prob(T) Mean Variance Parameter B2 Unit Size Estimate SE T Prob(T) Mean Variance

20 Dr. Wolfgang Langer - Methoden V: Grundlagen der Mehrebenenanalyse - WiSe 2000/ Parameter SIGMA Unit Size Estimate SE T Prob(T) Mean Variance Note: random level-1 coefficients are also referred to as level-2 outcomes See documentation for further elaboration on this subject MLA (R) Multilevel Analysis for Two Level Data Version Copyright Leiden University All Rights Reserved Part 5 Wed Oct 11 14:07: MLA version 3.2: annotated example Ordinary least squares estimates Fixed parameters Label Estimate SE G G G G Random parameters Label Estimate SE E(1) U1*U U2*U U2*U E(2) E(1): one-step estimate of sigma squared (ignoring grouping) E(2): two-step estimate of sigma squared See documentation for further elaboration on these subjects

21 Dr. Wolfgang Langer - Methoden V: Grundlagen der Mehrebenenanalyse - WiSe 2000/ MLA (R) Multilevel Analysis for Two Level Data Version Copyright Leiden University All Rights Reserved Part 6 Wed Oct 11 14:07: MLA version 3.2: annotated example Full information maximum likelihood estimates (BFGS) Fixed parameters Label Estimate SE T Prob(T) G G G G Random parameters Label Estimate SE T Prob(T) U1*U U2*U U2*U E Conditional intra-class correlation = 0.47/( ) = # iterations = 10-2*Log(L) = MLA (R) Multilevel Analysis for Two Level Data Version Copyright Leiden University All Rights Reserved Part 7 Wed Oct 11 14:07: MLA version 3.2: annotated example Residuals Level-2 residuals U1 Unit Raw Shrunken Mean Variance Covariance

22 Dr. Wolfgang Langer - Methoden V: Grundlagen der Mehrebenenanalyse - WiSe 2000/ K-S Z Prob(Z) Level-2 residuals U2 Unit Raw Shrunken Mean Variance Covariance K-S Z Prob(Z) Note: shrunken level-2 residuals are also referred to as conditional means Note: covariance refers to covariance with corresponding level-1 residuals See documentation for further elaboration on this subject MLA (R) Multilevel Analysis for Two Level Data Version Copyright Leiden University All Rights Reserved Part 8 Wed Oct 11 14:07: MLA version 3.2: annotated example Posterior means Parameter B1 Unit Estimate Mean Variance

23 Dr. Wolfgang Langer - Methoden V: Grundlagen der Mehrebenenanalyse - WiSe 2000/ Parameter B2 Unit Estimate Mean Variance Note: posterior means = shrunken estimates of random level-1 coefficients See documentation for further elaboration on this subject MLA (R) Multilevel Analysis for Two Level Data Version Copyright Leiden University All Rights Reserved Part 9 Wed Oct 11 14:07: MLA version 3.2: annotated example Diagnostics Level-2 sample size = 15 Total sample size = 230 Mean Level-1 sample size = 15 Effective sample size = 44 Squared correlation coefficients Norm based R-squared = Grand mean based R-squared = Context mean based R-squared = Trimmed mean based R-squared = Level-1 outliers (sorted by Prob) Level-1 Level-2 Level-1 Unit Unit Unit T Prob

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27 Dr. Wolfgang Langer - Methoden V: Grundlagen der Mehrebenenanalyse - WiSe 2000/ Level-2 Mahalanobis distances (sorted by Prob(M)) Unit M Prob(M) Effective sample size: N/(1+(N/J-1)*intra-class correlation) Squared correlation coefficients (R-squared) are highly speculative in nature Prob(M): probability - area under the curve of the chi-square distribution See documentation for further elaboration on this subject MLA (R) Multilevel Analysis for Two Level Data Version Copyright Leiden University All Rights Reserved Part 10 Wed Oct 11 14:07: MLA version 3.2: annotated example Bootstrap estimates (unbalanced unlinked shrunken residuals) Replications done = 200 Replications used = 200