XLSTAT-PLS
Contents
1. 1 Regression line this chart is displayed only if there is one explanatory variable and if that variable is quantitative 2 Explanatory variable versus standardized residuals this chart is displayed only if there is one explanatory variable and if that variable is quantitative 3 Dependent variable versus standardized residuals 4 Predictions versus observed values b Bar chart of the standardized residuals 23 Confidence intervals activate this option to display the confidence intervals on charts 1 and 4 Options for the PLS regression and the PCR Correlation charts activate this option to display the charts involving correlations between components and input variables In the case of PCR activate this option to display des correlation circle e Vectors activate this option to display the input variables with vectors Observations charts activate this option to display the charts that allow visualizing the observations in the new space e Labels activate this option to display the observations labels on the charts The number of labels can be modulated using the filtering option Biplots activate this option to display the charts where the input variables and the observations are simultaneously displayed e Vectors activate this option to display the input variables with vectors e Labels activate this option to display the observations labels on the biplots The number of labels can be modulated using2. User s manual XLSTAT PLS Copyright 2005 Addinsoft http www addinsoft com Table of Contents INTRODUCTION 1 INSTALLATION ssnsnsnsessnsnsnssssnsnsnssnensnensenssenensensnsnensnsensnenenssnsnenssnensnenenssnsnsnsnssnsnenssnensnenssssssnsnsnssnsnsnenssnsnensnssssnennnen 3 SYSTEM CONFIGURA TION sei idee de Pe HR TUE P ee PU etr e AN Nr e ee 3 INSTALLATION LICENSE cede HE ete och USING XLSTAT PLS smeeeeennenennennnnennnnnennnennnnnnnnnnnnnsnnesense 7 THEXESTAT PES APPROACH u ana gov t data esteem eee ete eere NE 7 DATA SELECTION cM S 7 MESSAGES P PE ne 9 OPTIONS vt eo Rte qnie en NI ti eet m omine et 9 PLS PCR OLS REGRESSION ususesesensnonsenensnsnsensnenensensnsnensensnsnenssnensnensenensnensnssnsnensnssssnennsensnsnenssnsnsnenssnssenensenene 13 DESCRIPTION M 13 DIADOG BOX 2 55 s e a inr UNUS 16 RESULTS is m eu ees e EXAMP ES T ANT a ere rites cop bes ns ln einen aides a red sehe lei Sirenen 30 REFERENCES 46 RR RP ERR eO Or nee nn RM REDE 30 Introduction XLSTAT PLS has been developed in order to make it possible for everyone to use PLS regression Partial Least Squares regression a modeling method that is more and more used in more and more domains The research of several Scandinavian scientists notably Wold and Martens at the beginning of the eighties made possible the emergence of this method that
3. the software To display the options dialog box of XLSTAT PLS click on Options in the menu or on the F button of the XLSTAT PLS toolbar o click this button to save the changes you have made ose click this button to close the dialog box If you haven t previously saved the options the changes you have made will not be kept te click this button to display the help ol click this button to reload the default options General tab Language use this option to change the language of the interface of XLSTAT PLS Dialog box entries e Memorize during one session activate this option if you want that XLSTAT PLS memorizes during one cession from opening until closing of XLSTAT PLS the entries and options of the dialog boxes e Memorize from one session to the next activate this option if you want that XLSTAT PLS memorizes the entries and options of the dialog boxes from one session to the next Included for data selections activate this option so that XLSTAT records the data selections from one session to the next This option is useful and saves your time if you work on spreadsheets that always have the same layout Ask for selections confirmation activate this option so that XLSTAT prompts you to confirm the data selections once you clicked on the OK button If you activate this option you will be able to verify the number of rows and columns of all the active selections Outputs tab Position of new sheet
4. and observed values for an ideal model the all the points would be on the bisecting line e the bar chart of the residuals 25 XLSTAT PLS If you have selected data to use in prediction mode a table displays the predictions on the new observations and the corresponding confidence intervals Results of the PCR regression The PCR regression requires a Principal Component Analysis step The first results concern the latter Eigenvalues the table of the eigenvalues and the corresponding scree plot are displayed The number of eigenvalues displayed is equal to the number of non null eigenvalues If a components filtering option has been selected it is applied only before the regression step If the corresponding outputs options have been activated XLSTAT PLS displays the factor loadings the coordinates of the input variables in the new space then the correlations between the input variables and the components The correlations are equal to the factor loadings if the PCA is performed on the correlation matrix The next table displays the factor scores the coordinates of the observations in the new space and are later used for the regression step If some observations have been selected for the validation they are displayed in this table A biplot is displayed if the corresponding option has been activated If the filtering option based on the correlations with the dependent variables has been selected the components used in the r
5. has proven to be very useful particularly when there are many variables up to thousands and when this number is even higher than the number of observations Such cases are met in the manufacturing industry when the quality of a product depends on many recorded parameters in chemistry when samples are described by wavelengths or the presence of many molecules as well as in marketing when the preference for a few products are given judges and described by many technical or sensory descriptors When the number of explanatory variables is greater than the number of variables the classical linear regression also named Ordinary Least Squares OLS regression cannot be used unless if you use a suitable variable selection method or if before running the regression you compute and select factors using a Principal Components Analysis this method is called Principal Components Regression PCR The reason why the OLS fails in that case is the multicollinearity between the explanatory variables that leads to numerical problems Furthermore the algorithms of PLS regression implemented in XLSTAT PLS allow handling properly the observations with missing values A preliminary estimation of the missing values is not necessary as it is with the OLS and PCR regressions Installation System configuration XLSTAT PLS runs under the following operating systems Windows 95 Windows 98 Windows Me Windows NT Windows 2000 and Windows XP A Mac OSX
6. if you do not want to include column E in the selection you should first select columns B to D with the mouse then press the Ctrl key and then select columns F to G still pressing Ctrl You may also select columns B to G then press Ctrl then select column E e Multiple selections with selection by rows cannot be used if the transposition option is not activated y button XLSTAT PLS e Multiple selections with selection by columns cannot be used if the transposition is activated button e When selecting a variable or a group of variables for example the quantitative explanatory variables you cannot mix the selection mode However you may use different modes for different selections within a dialog box e f you selected the name of the variables within the data selection you should make sure the Columns labels or Labels included option activated e You can use keyboard shortcuts to quickly select data Notice this is possible only you installed the latest patches for Microsoft Excel Here is a list of the most useful selection shortcuts 12 13 14 Ctrl A selects the whole spreadsheet Ctrl Space selects the whole column corresponding to the already selected cells Shift Space selects the whole row corresponding to the already selected cells When one or more cells are selected Shift Down selects the currently selected cells and the cells on the row below on one row Shift Up selects the curr
7. is not the case with Q cum The next table corresponds to the correlation matrix of the explanatory and dependent variables with the t and u components A chart displays the correlations with the t components The next table displays the w vectors followed by the w vectors and the c vectors that are directly involved in the model as it is shown in the Description section If to h 2 corresponds a valid model it is shown that the projection of the x vectors on the y vectors on the variables 23 XLSTAT PLS on the w c axes chart gives a fair idea of the sign and the relative weight of the corresponding coefficients in the model The next table displays the scores of the observations in the space of the t components The corresponding chart is displayed If some observations have been selected for the validation they are displayed on the chart The next table displays the standardized scoresof the observations in the space of the t components These scores are equivalent to computing the correlations of each observation represented by an indicator variable with the components This allows displaying the observations on the correlations map that follows where the Xs the Ys and the observations are simultaneously displayed An example of an an interpretation of this map is available in Tenenhaus 2003 The next table corresponds to the scores of the observations in the space of the u and then the u components The chart based on the
8. the cases where there a lot of explanatory variables other methods have been developed PCR Regression PCR Principal Components Regression can be divided into three steps we first run a PCA Principal Components Analysis on the table of the explanatory variables then we run an OLS regression on the selected components the we compute the parameters of the model that correspond to the input variables PCA allows to transform an X table with n observations described by variables into an S table with n scores described by q components where q is lower or equal to p and such that S S is invertible An additional selection can be applied on the components so that only the r components that are the most correlated with the Y variable are kept for the OLS regression step We then obtain the R table The OLS regression is performed on the Y and R tables In order to circumvent the interpretation problem with the parameters obtained from the regression XLSTAT PLS transforms the results back into the initial space to obtain the parameters and the confidence intervals that correspond to the input variables Using XLSTAT PLS PLS Regression This method is quick efficient and optimal for a criterion based on covariances It is recommended in cases where the number of variables is high and where it is likely that the explanatory variables are correlated The idea of PLS regression is to create starting from a table with n observations des
9. the residuals is accepted or rejected Cp the Mallows Cp is defined by p SEE cay Oo where SSE is the sum of squares of errors for the model with p explanatory variables and where G corresponds to the estimator of the variance of the residuals for the model that includes all the explanatory variables The closer the Cp coefficient to p the less biased the model AIC the Akaike s Information Criterion AIC is defined by 27 XLSTAT PLS CE ac win 5 Jer This criterion suggested by Akaike 1973 derives from the information theory and is based on the Kullback and Leibler measure 1951 It is a models selection criterion that penalizes models for which the addition of a new explanatory variable does not bring sufficient information The lower the AIC the better the model e SBC the Schwarz s Bayesian Criterion writes E ssc Wwin STF Jem w p This criterion suggested by Schwarz 1978 is close to the AIC and the goal is to minimize it e PC the Anemiya s Prediction Criterion writes 1 R W p W p PC This criterion suggested by Anemiya 1980 allows as the adjusted R to take into account the parsimony of the model e Press RMCE la Press RMSE statistic is displayed only if the corresponding option has been activated in the dialog box The Press statistic is defined by Press y W gt Den i l where Vics is the prediction of the i observation when it is not included in the data
10. u is displayed If some observations have been selected for the validation they are displayed on the chart The table with the Q quality indexes allows visualizing how the components contribute to the explanation of the dependent variables The table of the cumulated Q quality indexes allows measuring the quality that corresponds to a space with an increasing number of dimensions The table of the R and redundancies between the input variables dependent and explanatory and the components t and u allow evaluating the explanatory power of the t and u The redundancy between an X table n rows and p variables and a c component is the part of the variance of X explained by c We define it as the mean of the squares of the correlation coefficients between the variables and the component P Rd X c LY Rac P ja From the redundancies one can deduce the VIPs Variable Importance for the Projection that measure the importance of an explanatory variable for the building of the t components The VIP for the jth explanatory variable and the component h is defined by On the VIP charts one bar chart per component a border line is plotted to identify the VIPs that are greater than 0 8 this threshold suggested by Wold 1994 allows identifying the variables that contribute significantly The next table displays the outliers analysis The DModX distances from each observation to the model in the space of the X variables allow identifing t
11. variables in the model To compute the confidence intervals in the case of PLS regression the classical formulae based on the normality hypotheses used in OLS regression do not apply A bootstrap method suggested by Tenenhaus et al 2004 allows estimating the confidence intervals The greater the absolute value of a coefficient the greater the weight of the variable in the model When the confidence interval around the standardized coefficients includes 0 which can easily be observed on the chart the weight of the variable in the model is not significant In the predictions and residualstable the weight the observed value of the dependent variable the corresponding prediction the residuals and the confidence intervals are displayed for each observation Two types of confidence intervals are displayed an interval around the mean it corresponds to the case where the prediction is made for an infinite number of observations with a give set of values of the explanatory variables and an interval around an individual prediction it corresponds to the case where the prediction is made for only one observation The second interval is always wider than the first one as the uncertainty is of course higher If some observations have been selected for the validation they are displayed in this table The three charts that are displayed afterwards allow visualizing e the residuals versus the dependent variable e the distance between the predicted
12. you consent to the jurisdiction of such courts If any provision of this Agreement shall be invalid the validity of the remaining provisions of this Agreement shall not be affected This Agreement is the entire and Installation exclusive agreement between Addinsoft and you with respect to the Software and supersedes all prior agreements whether written or oral and other communications between Addinsoft and you with respect to the Software COPYRIGHT c 2004 BY Addinsoft SARL Paris FRANCE ALL RIGHTS RESERVED XLSTAT r IS A REGISTERED TRADEMARK OF Addinsoft SARL Paris FRANCE January 2005 Using XLSTAT PLS The XLSTAT PLS approach As all modules of the XLSTAT software suite XLSTAT PLS interface totally relies on Microsoft Excel whether for inputting the data or for displaying the results On the opposite computations are completely independent of Excel and the corresponding programs have been developed with the C programming language In order to guarantee irreproachable results the XLSTAT PLS software has been intensively tested and it has been validated by specialists of the statistical methods of interest Addinsoft has always been concerned about permanently improving the XLSTAT software suite and is welcoming the remarks and improvements you might want to suggest To contact Addinsoft write to supportOxIstat com Data selection As with all XLSTAT modules the selecting of data needs to be done directly on an Exc
13. Ew y 8 Le a With y Y wy R 1 n nt Y Gi i i 1 26 Using XLSTAT PLS The R is interpreted is the proportion of variability of the dependent variable explained by the model The close the R to 1 the better fitted the model The major drawback of the R is that it does not take into account the number variables used to fit the model Adjusted R the adjusted coefficient of determination of the model The adjusted R can be negative if the R is close to zero This coefficient is displayed only if of the model has not been fixed by the user The value of this coefficient is computed as follows R2 1 1 R2 p The adjusted R is a correction of the R that allows taking into account the number of variables used in the model MSE the Mean Squares of Errors MSE is defined by MSE Y w s y WS pr RMSE the Root Mean Squares of Errors RMSE is the square root of the MSE MAPE the Mean Absolute Percentage Error MAPE is computed as follows 100 MAPE Va de W Y i i 1 DW the Durbin Watson statistic is defined by n Yl 23 Maya DW Im x 3 y This statistic corresponds to the order 1 autocorrelation coefficient and allows verifying if the residuals are not autocorrelated The independence of the residuals is one of the hypotheses of the linear regression The user will need to look at a Durbin Watson table to know if the hypothesis of independence between
14. Leibler R A 1951 On information and sufficiency Annals of Mathematical Statistics 22 pp 79 86 Schwarz G 1978 Estimating the Dimension of a Model Annals of Statistics 6 pp 461 464 Tenenhaus M 1998 La Regression PLS Th orie et Pratique Technip Paris Tenenhaus M Pag s J Ambroisine L and amp Guinot C 2005 PLS methodology for studying relationships between hedonic judgements and product characteristics Food Quality an Preference 16 4 pp 315 325 Wold S 1995 PLS for multivariate linear modelling In van de Waterbeemd H ed QSAR Chemometric Methods in Molecular Design Vol 2 Weinheim Germany Wiley VCH pp 195 218 30
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16. The three methods give the same results if the number of components obtained from the PCA in PCR or from the PLS regression is equal to the number of explanatory variables 2 The components obtained from the PLS regression are built so that they explain as well as possible Y while the components of the PCR are built to describe X as well as possible XLSTAT PLS allows partly compensating this drawback of the PCR by allowing the selection of the components that are the most correlated with Y 15 XLSTAT PLS Dialog box The dialog is divided into several tabs that correspond to a variety of options ranging from the selection of data to the display of results You will find below the description of the various elements of the dialog box o click this button to start the computations Cancel j f click this button to close the dialog box without doing any computation oe click this button to display the help ol click this button to reload the default options IF click this button to delete the data selections click these buttons to change the way XLSTAT handles the data If the arrow points down XLSTAT considers that rows correspond to observations and columns to variables If the arrow points to the right XLSTAT considers that rows correspond to variables and columns to observations General tab Dependent variable s select the dependent variable s The data must be numerical If the Variable la
17. ame order in the selections On the other hand variable labels must not be selected the first row of the selections listed below must correspond to data Quantitative variable s activate this option to select the quantitative explanatory variables The first row must not include variable labels Qualitative Variable s activate this option to select the qualitative explanatory variables The first row must not include variable labels Observations labels activate this option if observations labels are available Then select the corresponding data If this option is not activated the observations labels are automatically generated by XLSTAT PLS PredObs1 PredObs2 Missing data tab These options are available only for PCR and OLS regression With PLS regression the missing data are automatically handled by the algorithm Remove observations activate this option to remove the observations with missing data Estimate missing data activate this option to estimate missing data before starting the computations e Mean or mode activate this option to estimate missing data by using the mean quantitative variables or the mode qualitative variables of the corresponding variables e Nearest neighbor activate this option to estimate the missing data of an observation by searching for the nearest neighbor of the observation Outputs tab Options common to the three methods Descriptive statistics activate this option
18. bels option is activated make sure that the headers of the variables have also been selected Quantitative variable s activate this option if you want to include one or more quantitative explanatory variables Then select the corresponding data The data must be numerical If the Variable labels option is activated make sure that the headers of the variables have also been selected Qualitative variable s activate this option if you want to include one or more qualitative explanatory variables Then select the corresponding data Whatever their Excel format the data are considered as categorical If the Variable labels option is activated make sure that the headers of the variables have also been selected Weights activate this option if you want to weight the observations If you do not activate this option the weights are considered to be equal to 1 The weights must be greater or equal to 0 If the Variable labels option is activated make sure that the header of the selection has also been selected Range activate this option if you want to display the results starting from a cell in an existing worksheet Then select the corresponding cell Sheet activate this option to display the results in a new worksheet of the active workbook Using XLSTAT PLS Workbook activate this option to display the options in a new workbook Variable labels activate this option if the first row of the data selections dependent and explanato
19. bles In this section Description Dialog box Results Example References Description The three regression methods available in this module have the common characteristic of generating models that involve linear combines of explanatory variables The difference between the three method lies on the way the correlation structures between the variables are handled OLS Regression From the three methods it is the most classical Ordinary Least Squares regression OLS is more commonly named linear regression simple or multiple depending on the number of explanatory variables In the case of a model with p explanatory variables the OLS regression model writes p Y B B X e jel where Y is the dependent variable Bo is the intercept of the model X corresponds to the a explanatory variable of the model 1 to p and is the random error with expectation 0 and variance o In the case where there are n observations the estimation of the predicted value of the dependent variable Y for the i observation is given by The OLS method corresponds to minimizing the sum of square differences between the observed and predicted values This minimization leads to the following estimators of the parameters of the model 13 XLSTAT PLS X DX X Dy 1 z dia omes rut m i l where is the vector of the estimators of the D parameters X is the matrix of the explanatory variables preceded by a vector of 1s y is t
20. cribed by p variables a set of h components with h p The method used to build the components differs from PCA and presents the advantage of handling missing data The determination of the number of components to keep is usually based on a criterion that involves a cross validation The user may also set the number of components to use Some programs differentiate PLS1 from PLS2 PLS1 corresponds to the case where there is only one dependent variable PLS2 corresponds to the case where there are several dependent variables The algorithms used by XLSTAT PLS are such that the PLS1 is only a particular case of PLS2 In the case of the OLS and PCR methods if models need to be computed for several dependent variables the computation of the models is simply a loop on the columns of the dependent variables table Y In the case of PLS regression the covariance structure of Y also influences the computations The equation of the PLS regression model writes Y T C E XW C T E 1 XW 5W C E where Y is the matrix of the dependent variables X is the matrix of the explanatory variables Th Ch Wh Wn et Ph are the matrices generated by the PLS algorithm and E is the matrix of the residuals The matrix B of the regression coefficients of Y on X with h components generated by the PLS regression algorithm is given by B W PW C h Note the PLS regression leads to a linear model as the OLS and PCR do Notes 1
21. egression step are those that have the greatest determination coefficients R with the dependent variables The matrix of the correlation coefficients between the components and the dependent variablesis displayed The number of components that are kept depends on the number of eigenvalues and on the selected options Min or Max components If the filtering option based on the eigenvalues has been selected the components used in the regression step are those that have the greatest eigenvalues The number of components that are kept depends on the number of eigenvalues and on the selected options 96 Min or Max components Results common to the PCR et OLS regressions Goodness of fit statistics this table displays statistics that are related to the goodness of fit of the regression model e Observations the number of observations taken into account for the computations In the formulae below n corresponds to number of observations e Sum of weights the sum of weights of the observations taken into account In the formulae below W corresponds to the sum of weights e DF the number of degrees of freedom of the selected model corresponds to the error DF of the analysis of variance table e R2 the coefficient of determination of the model This coefficient which value is between 0 and 1 is displayed only if the intercept of the model has not been fixed by the user The value of this coefficient is computed as follows
22. el sheet preferably with the mouse Statistical programs usually require that you first build a list of variables then define their type and at last select the variables of interest for the method you want to apply to them The XLSTAT approach is completely different as you only need to select the data directly on one or more Excel sheets Three selection modes are available e Selection by range you select with the mouse on the Excel sheet all the cells of the table that corresponds to the selection field of the dialog box e Selection by columns this mode is faster but requires that your data set starts on the first row of the Excel sheet If this requirement is fulfilled you may select data by clicking on the name A B of the first column of your data set on the Excel sheet and then by selecting the next columns by leaving the mouse button pressed and dragging the mouse cursor over the columns to select e Selection by rows this mode is the reciprocal of the selection by rows model It requires that your data set starts on the first column A of the Excel sheet If this requirement is fulfilled you may select data by clicking on the name 1 2 of the first row of your data set on the Excel sheet and then by selecting the next rows by leaving the mouse button pressed and dragging the mouse cursor over the rows to select Notes e Doing multiple selections is possible if your variables go from column B to column G and
23. ently selected and the cells on the row below on one row Shift Left selects the currently selected and the cells to the left on one column Shift Right selects the currently selected and the cells to the right on one column Crtl Shift Down selects all the adjacent non empty cells below the currently selected cells Crtl Shift Up selects all the adjacent non empty cells above the currently selected cells Crtl Shift Left selects all the adjacent non empty cells to the left of the currently selected cells Crtl Shift Right selects all the adjacent non empty cells to the right of the currently selected cells When one ore more columns are selected Shift Left selects one more column to the left of the currently selected columns Shift Right selects one more column to the right of the currently selected columns Crtl Shift Left selects all the adjacent non empty columns to the left of the currently selected columns Crtl Shift Right selects all the adjacent non empty columns to the right of the currently selected columns When one or more rows are selected Shift Down selects one more row to the left of the currently selected rows Using XLSTAT PLS 17 Shift Up selects one more row to the right of the currently selected rows 18 Crtl Shift Down selects all the adjacent non empty rows to the left of the currently selected rows 19 Crtl Shift Up selects all the adjacent non empty rows to the right of the currently
24. ents Options for PCR and OLS regression Intercept activate this option to fix the intercept or constant of the model to a given value Then enter the value in the corresponding field 0 by default Tolerance activate this option to allow the OLS algorithm to automatically remove the variables that would either be constant or highly correlated with other variables or group of variables Minimum and default value is 0 0001 Maximum value allowed is 1 The higher the tolerance the more the model tolerates collinearities between the variables Options for OLS regression Constraints this option is active only if you have selected qualitative explanatory variables Choose the type of constraint e al 0 for each qualitative variable the parameter of the model that corresponds to the first category of the variable is set to 0 This type of constraint is useful when you consider that the first category corresponds to a standard or to a null effect e Sum ai 0 for each qualitative variable the sum of the parameters corresponding to the various categories equals 0 e Sum ni ai 0 for each qualitative variable the sum of the parameters corresponding to the various categories weighted by their frequency equals 0 Model selection activate this option if you want to use one of the following model selection methods e Best model this method allows choosing the best model among all the models that are based on a number of var
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26. he outliers for the explanatory explicatives while the DModY distances from each observation to the model in the space of the Y variables allow identifying the outliers for the dependent variables On the corresponding charts the threshold values DCrit are also displayed to help identying of the outliers the DMod values that are above the DCrit threshold correspond to outliers The DCrit are computed using the threshold values classically used in box plots The value of the DModX for the i observation writes 24 Using XLSTAT PLS P Y ex j l p h where the e X t i 1 n are the residuals of the regression of X on the D component The value of the DModY for the it observation writes where q is the number of dependent variables and the e Y t i 1 n are the residuals of the regression of Y on the j component The next table displays the parameters of the models corresponding to the one ar more dependent variables It is followed by the equation corresponding to each model if the number of explanatory variables does not exceed 20 For each of the dependent variables a series of tables and charts is displayed Goodness of fit statistics this table displays the goodness of fit statistics of the PLS regression model for each dependent variable The definition of the statistics is as follows The table of the standardized coefficients also named beta coefficients allows comparing the relative weight of the
27. he vector of the n observed values of the dependent variable p is the number of explanatory variables to which we add 1 if the intercept is not fixed wis the weight of the i observation and Wis the sum of the w weights and Dis a matrix with the w weights on its diagonal The vector of the predicted values writes X X DX X Dy The limitations of the OLS regression come from the constraint of the inversion of the X X matrix it is required that the rank of the matrix is p 1 and some numerical problems may arise if the matrix is not well behaved XLSTAT PLS uses algorithms due to Dempster 1969 that allow circumventing these two issues if the matrix rank equals q where qis strictly lower than p 1 some variables are removed from the model either because they are constant or because they belong to a block of collinear variables Furthermore an automatic selection of the variables is performed if the user selects a too high number of variables compared to the number of observations The theoretical limit is n 1 as with greater values the X X matrix becomes non invertible The deleting of some of the variables may however not be optimal in some cases we might not add a variable to the model because it is almost collinear to some other variables or to a block of variables but it might be that it would be more relevant to remove a variable that is already in the model and to the new variable For that reason and also in order to handle
28. iables that is bounded by Min variables and Max variables The quality of the model depends on a selection Criterion Using XLSTAT PLS 20 Criterion select the criterion in the following list adjusted R Mean Squares of Errors MSE Mallows Cp Akaike s AIC Schwarz s SBC Anemiya s PC 21 Min variables enter the minimum number of variables to take into account in the model 22 Max variables enter the maximum number of variables to take into account in the model Note this method can lead to very long computations because the total number of models explored is the sum of the Cn k where k varies between Min variables and Max variables and where Cn k is n n k k It is therefore highly recommended that you increase step by step the value of Max variables Stepwise the selection process starts with the adding of the variable that contributes the most to the model the criterion used here is the Student s t statistic If a second variable is such that the probability of its t is lower than the Threshold level it is added to the model The procedure is the same for the third variable Then starting with the third variable the algorithm evaluates how the removal of one of the variables would impact the model If the probability corresponding to the Student s t of one of the variables is greater than the Threshold level the variable is removed The procedure continues until no variable can be either added or rem
29. ined below e User defined activate this option if you want that XLSTAT displays charts with dimensions as defined by the following values Width enter the value in points of the charts width Height enter the value in points of the charts height Display orthonormal charts activate this option to display orthonormal charts when this is relevant Displaying orthonormal allows making sure that there is no distortion effect due to different scales of the abscissa and ordinates axes that could lead to misinterpretations Advanced tab Random numbers Fix the seed to activate this option if want to make sure that the computations involving random numbers always give the same result Then enter the seed value Path for the user s files this path can be modified if and only if you have administrator rights on the machine You can then modify the folder where the user s files are saved by clicking the button that will display a box where you can select the appropriate folder User s files include the general options as well as the options and selections of the dialog boxes of the various XLSTAT functions The folder where the user s files are stored must be accessible for reading and writing to all types of users 11 PLS PCR OLS Regression Use this module to model and predict the values of one or more dependant quantitative variables using a linear combination of one or more explanatory quantitative and or qualitative varia
30. natory variables among the dependent variables and between both groups Results of the PLS regression The first table displays the model quality indexes The quality corresponds here to the cumulated contribution of the components to the indexes e The Q cum index measures the global contribution of the h first components to the predictive quality of the model and of the sub models if there are several dependent variables The Q cum h index writes Y PRESS Q cum h 1 E 2 SCE The index involves the PRESS statistic that requires a cross validation and the Sum of Squares of Errors SSE for a model with one less component The search for the maximum of the Q cum index is equivalent to finding the most stable model e The R Ycum index is the sum of the coefficients of determination between the dependent variables and the h first components It is therefore a measure of the explanatory power of the first components for the dependent variables of the model e The R Xcum index is the sum of the coefficients of determination between the explanatory variables and the h first components It is therefore a measure of the explanatory power of the h first components for the explanatory variables of the model A bar chart is displayed to allow the visualization of the evolution of the three indexes when the number of components increases While the R2Ycum and R Xcum indexes necessarily increase with the number of components this
31. ndardized coefficients also named beta coefficients allows comparing the relative weight of the variables in the model The greater the absolute value of a coefficient the greater the weight of the variable in the model When the confidence interval around the standardized coefficients includes 0 which can easily be observed on the chart the weight of the variable in the model is not significant In the predictions and residualstable the weight the value of the explanatory variable if there is only one the observed value of the dependent variable the corresponding prediction the residuals and the confidence intervals the adjusted prediction and the Cook s D are displayed for each observation Two types of confidence intervals are displayed an interval around the mean it corresponds to the case where the prediction is made for an infinite number of observations with a give set of values of the explanatory variables and an interval around an individual prediction it corresponds to the case where the prediction is made for only one observation The second interval is always wider than the first one as the uncertainty is of course higher If some observations have been selected for the validation they are displayed in this table The charts that follow allow visualizing the results listed above If there is only one explanatory variable in the model and if that variable is quantitative then the first chart allows visualizing the observati
32. ons the regression line and the confidence intervals around the prediction The second chart displays the standardized residuals versus the explanatory variable The residuals should be randomly distributed around the abscissa axis If a trend can be observed that means there is a problem with the model The three charts that are displayed afterwards allow visualizing respectively the standardized residuals versus the dependent variable the distance between the predicted and observed values for an ideal model the all the points would be on the bisecting line and the bar chart of the standardized residuals The third chart makes it possible to quickly see if there is an unexpected number of high residuals the normality assumption for the residuals is such that only 5 of the standardized residuals should be out of the 2 2 interval If you have selected data to use in prediction mode a table displays the predictions on the new observations and the corresponding confidence intervals OLS regression results If the Type SS and Type Ill SS SS Sum of Squares options have been activated the corresponding tables are displayed The Type I SS table allows visualizing the influence of the progressive addition of new explanatory variables to the model The influence is given by the Sum of Squares of Errors SSE de la Mean Squares of Errors MSE the Fisher s F statistic and the probability corresponding to the Fisher s F The smaller the
33. oved from the model Forward the procedure is identical to the stepwise except that there are no removal steps Backward the procedure starts with the selection of all the available variables The variables are then removed from the model one by one using the same methodology as for the stepwise selection Threshold level enter the value of the threshold probability for the Student s t statistic during the selection process Validation tab Validation activate this option if you want to use a sub sample of the data to validate the model Validation set choose one of the following options to define how to obtain the observations used for the validation Random the observations are randomly selected The Number of observations N must then be specified N last rows the N last observations are selected for the validation The Number of observations N must then be specified N first rows the N first observations are selected for the validation The Number of observations N must then be specified Group variable if you choose this option you need to select a binary variable with only Os and 1s The 1s identify the observations to use for the validation Prediction tab 19 XLSTAT PLS Prediction activate this option if you want to select data to use them in prediction mode If activate this option you need to make sure that the prediction dataset is structured as the estimation dataset same variables with the s
34. probability the more information the variable brings to the model Note the order of selection of the variables influences the results obtained here The Type III SS table allows visualizing the influence of the withdrawal of an explanatory variable on the goodness of fit of the model all the other variables being included The influence is measured by the Sum of Squares of Errors SSE de la Mean Squares of Errors MSE the Fisher s F statistic and the probability corresponding to the Fisher s F The smaller 29 XLSTAT PLS the probability the more information the variable brings to the model Note the order of the variables in the selection does not influence the results in this table Examples Tutorials on how to use XLSTAT PLS are available on the Addinsoft website on following pages http www xlstat com demo pls htm http www xIstat com demo per htm References Akaike H 1973 Information Theory and the Extension of the Maximum Likelihood Principle Second International Symposium on Information Theory V N Petrov and F Csaki Budapest Akailseoniai Kiudo pp 267 281 Anemiya T 1980 Selection of Regressors International Economic Review 21 pp 331 354 Bastien P Esposito Vinzi V and Tenenhaus M 2005 PLS Generalised Regression Computational Statistics and Data Analysis 48 pp 17 46 Dempster A P 1969 Elements of Continuous Multivariate Analysis Addison Wesley Reading MA Kullback S and
35. ry variables weights observations labels includes a header Observation labels activate this option if observations labels are available Then select the corresponding data If the Variable labels option is activated you need to include a header in the selection If this option is not activated the observations labels are automatically generated by XLSTAT PLS Obs1 Obs2 Method choose the regression method you want to use e PLS activate this option to compute a Partial Least Squares regression e PCR activate this option to compute Principal Components Regression e OLS activate this option to compute an Ordinary Least Squares regression Options tab Common options Confidence interval enter the size in of the confidence interval that is used for the various tests parameters and predictions Default value 95 Options for PLS regression Stop conditions e Automatic activate this option so that XLSTAT PLS automatically determines the number of components to keep e Qi threshold activate this option to fix the threshold value of the Qi criterion used to determine if the contribution of a component is significant or not The default value is 0 0975 which corresponds to 1 0 95 e Qi improvement activate this option to fix the threshold value of the Qi improvement criterion used to determine if the contribution of a component is significant or not The default value is 0 05 which corresponds to a 596 impro
36. s if you choose the Sheet option in the dialog boxes of the XLSTAT functions use this option to modify the position if the results sheets in the Excel workbook Number of decimals choose the number of decimals to display for the numerical results Notice that you always have the possibility to view a different number of decimals afterwards by using the Excel formatting options Display titles in bold activate this option so that XLSTAT displays the titles of the results tables in bold Using XLSTAT PLS Display table headers in bold activate this option to display the headers of the results tables in bold Display the results list in the report header activate this option so that XLSTAT displays the results list at the bottom of the report header Display the project name in the report header activate this option to display the name of your project in the report header Then enter the name of your project in the corresponding field Charts tab Display charts on separate sheets activate this option if you want that the charts are displayed on separate chart sheets Note when the charts are displayed on a spreadsheet you can still transform them into a chart sheet by clicking the right button of the mouse and then selecting location and then As new sheet Charts size e Automatic choose this option if you want that XLSTAT automatically determines the size of the charts using as a starting value the width and height def
37. selected rows See also http www xlstat com demo select htm Messages XLSTAT PLS uses an innovative message system to give information to the user and to report problems The dialog box below is an example of what happens when an optional selection field Quantitative variable s has been activated but left empty The software detects the problem and displays the message box XLSTAT Message SELEODOZ2 A Quantitative variable s The selection is empty Please select data or uncheck the corresponding option XLSTAT PLS 1 0 Excel 11 0 6113 Windows 32 bit NT http fvaww xlstat com cma The information displayed in red or in blue depending on the severity to indicate which object option selection is responsible for the message If you click on back the dialog box of the PLS regression will be displayed again and the field corresponding to the Quantitative variable s is activated This message should be explicit enough to help you solve the problem by yourself If a tutorial is available the hyperlink http www xlstat com links to a tutorial on the subject related to the problem Sometimes an email address is displayed below the hyperlink to allow you send an email to Addinsoft using your usual email software with the content of the XLSTAT message being automatically displayed in the email message Options XLSTAT PLS XLSTAT PLS offers several options in order to allow you to customize and optimize the use of
38. set used for the estimation of the parameters of the model When obtain Press W p Press RMCE The Press RMSE can then be compared to the RMSE A large difference between both indicates that the model is sensitive to the presence or absence of some observations The analysis of variance table allows evaluating how much information the explanatory variables bring to the model In the case where the intercept of the model is not fixed by the user the explanatory power is measured by comparing the fit of the selected model with the fit of a basic model where the dependent variable equals its mean When the intercept is fixed to a given value the selected model is compared to a basic model where the dependent model equals the fixed intercept In the case of a PCR regression the first table of model parameters corresponds to the parameters of the model based on the selected components This table is not easy to interpret For that reason a transformation is performed to obtain the parameters of the model corresponding to the input variables The latter table is directly obtained in the case of an OLS regression In this table you will find the estimate of the parameters the corresponding 28 Using XLSTAT PLS standard error the Student s t the corresponding probability as well as the confidence interval The equation of the model is then displayed to facilitate the visualization or the reuse of the model The table of the sta
39. the filtering option Colored labels activate this option to display the labels with the same color as the corresponding points If this option is not activated the labels are displayed in black Filter activate this option to modulate the number of labels displayed e Random the observations to display are randomly selected The Number of observations N to display must then be specified e N first rows les N first observations are displayed on the chart The Number of observations N to display must then be specified e Nast rows the N last observations are displayed on the chart The Number of observations N to display must then be specified e Group variable if you choose this option you need to select a binary variable with only Os and 1s The 1s identify the observations to display 22 Using XLSTAT PLS Results Descriptive statistics the tables of descriptive statistics display for all the selected variables a set of basic statistics For the dependent variables colored in blue and the quantitative explanatory variables XLSTAT PLS displays the number of observations the number of observations with missing data the number of observations with no missing data the mean and the unbiased standard deviation For the qualitative explanatory variables XLSTAT PLS displays the name and the frequency of the categories Correlation matrix this table is displayed to allow your visualizing the correlations among the expla
40. the outliers analysis Options for PCR regression Factor loadings activate this option to display the factor loadings The factor loadings are equal to the correlations between the principal components and the input variables if the PCA is based on the correlation matrix standardized PCA Correlations Factors Variables activate this option to display the correlations between the principal component and the input variables Factor scores activate this option to display the factor scores coordinates of the observations in the new space generated by the PCA The scores are used in the regression step of the PCR Options for PCR and OLS regression Analysis of variance activate this option to display the analysis of variance table Adjusted predictions activate this option to compute and display the adjusted predictions in the predictions and residuals table Cook s D activate this option to compute and display the Cook s distances in the predictions and residuals table Press activate this option to compute and display the Press statistic Charts tab Options common to the three methods Regression charts activate this option to display the regression charts e Standardized coefficients activate this option to display a chart with the standardized coefficients of the model and the corresponding confidence intervals 21 XLSTAT PLS e Predictions and residuals activate this option to display the following charts
41. to display the descriptive statistics for all the selected variables Correlations activate this option to display the correlation matrix for the quantitative variables dependent and explanatory Standardized coefficients activate this option to display the standardized parameters of the model also name beta coefficients Equation activate this option to explicitly display the equation of the model Predictions and residuals activate this option to display the table of predictions and residuals Option for PLS regression 20 Using XLSTAT PLS Bootstrap intervals activate this option to compute the confidence intervals of the standardized coefficients The computations involve a bootstrap method and can therefore slow down the computations depending on the number of observations The intervals are computed using the BC a intervals suggested in Bastien et al 2005 t u and u components activate this option to display the tables corresponding to the components If this option is not activated the corresponding charts are not displayed C w W and p vectors activate this option to display the tables corresponding to the vectors obtained from the PLS algorithm If this option is not activated the corresponding charts are not displayed VIPs activate this option to display the table and the charts of the Variable Importance for the Projection Outliers analysis activate this option to display the table and the charts of
42. uiring the Software Some jurisdictions do not allow the foregoing limitations of liability so the foregoing limitations may not apply to you 6 TERM AND TERMINATION This Agreement shall continue until terminated You may terminate the Agreement at any time by deleting all copies of the Software This license terminates automatically if you violate any terms of the Agreement Upon termination you must promptly delete all copies of the Software 7 CONTRACTING PARTIES If the Software is installed on computers owned by a corporation or other legal entity then this Agreement is formed by and between Addinsoft and such entity The individual executing this Agreement represents and warrants to Addinsoft that they have the authority to bind such entity to the terms and conditions of this Agreement 8 INDEMNITY You agree to defend and indemnify Addinsoft against all claims losses liabilities damages costs and expenses including attorney s fees which Addinsoft may incur in connection with your breach of this Agreement 9 GENERAL The Software is a commercial item This Agreement is governed and interpreted in accordance with the laws of the Court of Paris France without giving effect to its conflict of laws provisions The United Nations Convention on Contracts for the International Sale of Goods is expressly disclaimed Any claim arising out of or related to this Agreement must be brought exclusively in a court located in PARIS FRANCE and
43. vement This value is computed as follows Q h Q h 1 Q h Imp ime Gin e Minimum Press activate this option so that the number of components used in the model correspond to the model with the minimum Press statistic e Max components activate this option to set the pour fixer le maximum number of components to take into account in the model 17 XLSTAT PLS Options for PCR regression Standardized PCA activate this option to run a PCA on the correlation matrix Inactivate this option to run a PCA on the covariance matrix unstandardized PCA Filter components you may activate one of the two following options to reduce the number of components used in the model e Min activate this option and enter the minimum percentage of total variability that the selected components should represent e Max number activate this option to fix the maximum number of components to take into account Sort components by choose one of the following options to determine which criterion should be used to select the components on the basis of the Min 95 or of the Max Number e Correlations with Ys activate this option so that the components selection is based on the sorting down of R coefficient between the dependent variable Y and the components This option is recommended e Eigenvalues activate this option so that the selection of the components is based on the sorting down of the eigenvalues corresponding to the compon
44. will soon be available To be able to run XLSTAT PLS required that Microsoft Excel is also installed on your computer XLSTAT PLS is compatible with the following Excel versions Excel 97 8 0 Excel 2000 9 0 Excel XP 10 0 and Excel 2003 1 1 0 Free patches and upgrades for Microsoft Office are available for free on the Microsoft Website We highly recommend that you download and install these patches as some of them are critical To check if your Excel version is up to date please go from time to time to the following web site http office microsoft com officeupdate Installation To install XLSTAT PLS you need to e Either double click on the xlstatpls exe filed that you downloaded from the XLSTAT website www xlstat com or from one of our numerous partners or available on a CD Rom e Or insert the CD Rom you received from us or from a distributor and wait until the installation procedure starts and then follow the step by step instructions If your rights on your computer are restricted you should ask someone that has administrator rights on the machine to install the software for you Once the installation is over the administrator must let you have read and write access to the following folders and keys e Hard disk folder the folder where the XLSTAT user files are located typically C Documents and settings User Name Application Data Addinsoft XLSTAT PLS License XLSTAT PLS 1 0 SOFTWARE LICENSE AGREEMENT ADDINS
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