Internet Utilisation in 112 Countries. Robust Regression Diagnostics Project. Submitted by : xxxx xxxx. Submitted to: Prof. Ali S.

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1 Internet Utilisation in 112 Countries Robust Regression Diagnostics Project Submitted by : xxxx xxxx Submitted to: Prof. Ali S. Hadi Date: June, 20xx 1

2 Table of content 1. Introduction What question can be answered by the analysis of the data? Background or literature review Description of the data Source Explanation of the variables Initial model Hypothesis of the problem Data analysis and assumptions testing Graphical examination of the data and transformations Transformed model fit to the data Criticisms and graphs after fitting the model to the data Identification of unusual observations Repeating the above steps without the 6 detected influential measures Repeating the above steps without the 7 detected influential measures Summary and conclusions Final model Conclusions Appendix Annotated Computer outputs List of data set

3 Table of attachments Tables Table 1: Summary of the initial variables Table 2: Correlation matrix of the initial variables Table 3 : Summary of the transformed variables Table 4: Correlation matrix of the transformed variables Table 5: Model summary of the backward selection method (1 st fitting) Table 6: Detailed analysis of the backward model selection steps (1 st fitting) Table 7: Model summary of the stepwise selection method (1 st fitting) Table 8: Detailed analysis of the stepwise model selection steps (1 st fitting) Table 9: Identifed influencial measures (1 st fitting) Table 10: Model summary of the backward selection method (2 nd fitting) Table 11: Detailed analysis of the backward model selection steps (2 nd fitting) Table 12: Model summary of the stepwise selection method (2 nd fitting) Table 13: Detailed analysis of the stepwise model selection steps (2 nd fitting) Table 14: Stepwise selection method for data set without 7 influencial measures (3 rd fitting)

4 Figures Figure 1: Scatter plot INT vs GDP Figure 2: Scatter plot INT vs POP Figure 3: Scatter plot INT vs COMPUTER Figure 4: Scatter plot INT vs AREA Figure 5: Scatter plot INT vs CPI Figure 6: Scatter plot INT vs URBANPOP Figure 7: Scatter plot INT vs TELCOST Figure 8: Scatter plot INT vs MOBTEL Figure 9: Histogramm and normal probability plot of the transformed variable SQR_INT Figure 10: Histogramm and normal probability plot of the transformed variable Ln_GDP Figure 11: Histogramm and normal probability plot of the transformed variable Ln_POP Figure 12: Histogramm and normal probability plot of the transformed variable SQR_CRM Figure 13: Histogramm and normal probability plot of the transformed variable Ln_ARA Figure 14: Histogramm and normal probability plot of the transformed variable 1/CPI Figure 15: Histogramm and normal probability plot of the transformed variable SQR_MBL Figure 16: Histogramm and normal probability plot of the variable TELCOST Figure 17: Histogramm and normal probability plot of the variable URBAN Figure 18: Scatterplot matrix of the transformed variables Figure 19: Added-variable plot Sqr(Int) residuals vs Ln_GDP residuals (1 st fitting) Figure 20: Added-variable plot Sqr(Int) residuals vs Ln_POP residuals (1 st fitting) Figure 21: Added-variable plot Sqr(Int) residuals vs SQR_CMR residuals (1 st fitting) Figure 22: Added-variable plot Sqr(Int) residuals vs Ln_ARA residuals (1 st fitting) Figure 23: Added-variable plot Sqr(Int) residuals vs 1/ CPI residuals (1 st fitting) Figure 24: Added-variable plot Sqr(Int) residuals vs URP residuals (1 st fitting) Figure 25: Added-variable plot Sqr(Int) residuals vs SQR_MBL residuals (1 st fitting) Figure 26: Added-variable plot Sqr(Int) residuals vs TELCOST residuals (1 st fitting) Figure 27: Scatter plot SQR_INT vs LnCMR (1 st fitting) Figure 28: Scatter plot SQR_INT vs SQR_MBL (1 st fitting) Figure 29: Plots of studentized residuals versus SQR_CMR and versus SQR_MBL (1 st fitting) Figure 30: Plots of studentized residuals versus SQR_MBL (1 st fitting) Figure 31: Histogramm of the residuals (1 st fitting) Figure 32: Scatter plot of the studentized residuals vs the normal scores (1 st fitting) Figure 33: Plolt of residuals vs. sqr_cmr (2 nd fitting) Figure 34: Plot of residuals vs sqr_mbl (2 nd fitting) Figure 35: Normal probability plot fo the residuals (2 nd fitting) Figure 36: Histogramm of the residuals (2 nd fitting) Figure 37: Plot of residuals versus predicted values (2 nd fitting) Figure 38: Plot of residuals versus predictors (3 rd fitting) Figure 39: Normal probability plot and histogramm of the residuals (3 rd fitting) Figure 40: Scatter plot of residuals vs predicted values (3 rd fitting) Figure 41: Plot of leverage values versus the index (3 rd fitting) Figure 42: Plots for identification of influcencial obervations (3 rd fitting) Figure 43: Plot of Potential-Residuals (3 rd fitting) Figure 44: Leverage versus squared residual plot (3 rd fitting)

5 1. Introduction 1.1 What question can be answered by the analysis of the data? The purpose of this regression is to analyse the variables influencing the number of internet users per 1000 people in 112 countries. 1.2 Background or literature review Prof. A. Hadi, Course notes of Robust Regression Diagnostics, Postgrade of Statistics, University of Neuchatel, Switzerland, 2005 S. Chatterjee, A. Hadi, B. Price, Regression Analysis by Example, 3 rd ed., John Wiley & Sons, Inc., S. Chatterjee, A. Hadi, Sensitivity Analysis in Linear Regression, John Wiley & Sons, Description of the data 2.1 Source Explanation of the variables Dimension : 112 observations on 9 variables Description : The data set has been defined individually on the base of the two internet sites mentioned above. The data were collected for the years between 1999 and 2001, depending on the availability of the data. I chose at first all the possible countries and deleted the one with missing values. The purpose of this regression is to analyse the variables influencing the number of internet users per 1000 people in 112 countries. The variables are: INTERNET: Internet users per 1000 people (Internet users are people with access to the world-wide network) GDP: Gross domestic product (GDP) per capita (in USD) POP: Population (thousand people) COMPUTER: Number of personal computers per 1000 people AREA: Total area in thousand hectares 5

6 TELCOST: Telephone average cost of local call in USD per three minutes. (Cost of local call is the cost of a three-minute, peak rate, fixed line call within the same exchange area using the subscriber's equipment (that is, not from a public phone)). CPI: Consumer price index (CPI) base 1995 = 100 (Consumer price index reflects changes in the cost to the average consumer of acquiring a fixed basket of goods and services that may be fixed or changed at specified intervals) URBANPOP: Urban population in % of total. (Urban population is the share of the total population living in areas defined as urban in each country) MOBTEL: Number of mobile phones subscribers per 1000 people 2.3 Initial model The initial model according to the statement of the problem for explaining the number of internet user in 112 countries is the following: INTERNET = β 0 +β 1*GDP + β 2 *POP+ β 3*COMPUTER + β 4 * AREA + β 5*TELCOST + β 6*CPI + β 7*URBANPOP + β 8*MOBTEL 2.4 Hypothesis of the problem My hypothesis is that GDP, COMPUTER, URBANPOP and MOBTEL have a positive influence on INTERNET, that TELCOST and CPI have a negative influence on INTERNET. The sense of a relationship between POP, AREA and INTERNET is a priori not clear. The goal of this paper is to find out, which variables influence the number of internet user per 1000 people in the analysed countries and, in which way they influence it. 3. Data analysis and assumptions testing 3.1 Graphical examination of the data and transformations For the exploration of the data I start the analysis by summarizing the different values I want to use for the estimation of the model. (appendix, table 1). The analysis of the summary shows that the magnitudes of the standard deviations are very high for all the variables at the exception of one, which is TELCOST. For having a better understanding of the relationship between the response variable and the predictors, I do a scatter plot of the response variable versus each predictor (figure 1-8) as well as a correlation matrix of the variables (table 2). One observe on the scatter plots a relationship between INT and GDP, COMPUTER, MOBTEL as well as URBANPOP, which seems linear only between GDP, COMPUTER and MOBTEL. This relationship is confirmed by a high correlation between these variables (close to 0,9 or more for the 3 first variable and 0,6 6

7 for URBANPOP. In order to stabilise this standard deviation, to try to reach a linear relationship between the response variable and the predictors, and to approach a certain normality in the distribution of each variable, I proceed to the following transformations (see the histograms and the normal probability plots in figures 9 to 17 : => INT transformed in SQR_INT => GDP transformed in Ln_GDP => POP transformed in Ln_POP => CMR transformed in SQR_CMR => AREA transformed in Ln_AREA => CPI transformed in 1/CPI => MBL transformed in SQR_MBL => URBAN not transformed => TELCOST not tranformed A summary of the transformed variables as well as the correlation matrix is sown in tables 3 and 4 of the appendix. Bases on the transformation done, the new model to analyse is the following: (INTERNET) = β 0 +β 1*Ln(GDP) + β 2 *Ln(POP)+ β 3* (COMPUTER) + β 4 * Ln(AREA) + β 5*TELCOST + β 6*1/(CPI) + β 7*URBANPOP + β 8* (MOBTEL) Before starting the regression analysis, it is important to examine the relationship between the variables of the model by mean of a scatter plot matrix (figure 18). The scatter plot matrix is based on transformed variables. The focus is first on the top-line of the matrix because it shows the relationship between the response variable SQR_INT and all the predictors. The first line of matrix shows a linear trend between the response variable and LN_GDP, SQR_CMR, SQR_MBL and URP. The correlation matrix (table 4) proves this assumption. Any real relationship has been noticed between the response variable and the other predictors. A linear trend has also been observed between two groups of predictors: LN_GDP, SQR_CMR, URBANPOP,SQR_MBL and LN_POP, LN_ARA. The correlation matrix is also useful for investigating on multi-collinearity problems between the predictors and confirms a strong correlation between the predictors for which, a linear trend has been observed. This correlation between predictors has been mentioned, but it does not necessary mean that multi-collinearity problem will arise. 3.2 Transformed model fit to the data 7

8 The results of a backward and stepwise selection methods are shown in the appendix, tables 5-8. The resulting regression model is the same for the 2 methods computed automatically in SPSS. I tried manually the backward method and the resulting model confirmed the two models obtained automatically. The resulting model after fitting is the following: (INTERNET) = 0, ,622* (COMPUTER) + 0,357* (MOBTEL) Unfortunately it is too early for assessing of the quality of this model. It is important to perform a detailed control of the model by analysing every step of the backward and stepwise method selection for checking the existence of a possible multi-collinearity problem. The detailed analysis of table 6 and 8 of the different steps shows that basically no multi-collinearity problem arises during the backward and forward selection. Tables 6 and 8 confirms the absence of multi-collinearity within the two variables of the final model by showing the variance influence factors (VIF) lower than 10 for each of the variables. However I noticed a multi-collinearity problem for Ln_GDP (VIF higher than 10), but this variable has been excluded of the selected model at step 5 of the backward selection. 3.3 Criticisms and graphs after fitting the model to the data It is now important at this stage of the analysis to check if the t-values of the variables are really reliable or if they are influenced by potential hypothesis violations. This check can be done with the added-variable plot as complement of the t-test for the significance of the regression coefficient of the variable V (added variable). An added-variable plot is done for every co-variate of the initial model. Figures of the appendix show these plots. The analysis of these plots confirms the t-test results and shows significant linear relationships for the two predictors (COMPUTER) and (MOBTEL) retained in the selected model. The next step is to check the regression assumptions, which are namely (a) linearity of the model in β, (b) linear independence of the predictor variables, (c) distributional assumption of the X (non-random and measured without errors), distributional assumption of the error (normally distributed, independent, zero mean and constant variance), (d) implicit assumption (observations are equally reliable and have equal influence in determining the least squares results and the conclusions drawn from the results). Figures 29 to 30 in the appendix show the plots of the sudentized residuals versus each of the predictors selected in the model. On these plots one observes a quite random scatter of points. Two groups of points can be observed. One observes two groups, one which can represent the industrialised countries on the right 8

9 and on the left the non or less industrialised countries. It also shows the linearity of the predictors. The variance of the variables SQR_MBL and SQR_CMR looks homogenous within the groupes. It is now important to check the distribution assumption about the residuals to see if they follow a normal distribution. Figure 31 shows the histogram of the residuals. One observes a quite good normal approximation for most of the residuals but it shows also clearly the presence of some extreme points. The presence of extreme values observed on the histogram of means that the normality assumption is violated. Figure 32 normal probability plot of the residuals. If the normality assumption where hold, one should oberve a quite straight line, which is not the case. This means again that the norrmality assumption is violated. The next steps of this work is now to identify theses unusual observations and to analyse them. Without such an analysis, it is impossible to assess the quality of the selected model. 3.4 Identification of unusual observations A distinction has to be made between three types of unusual observations: the outliers, the high leverage points and the influential observations. As the quality of the selected model can not be assessed, it is important to analyse these three types of unusual observations Identification of outliers Graphical displays of residuals are useful for providing information about the presence of outliers. Let s begin our graphical analysis by the normal probability plot of the residuals. This is a scatter-plot of the studentized residuals versus the normal scores (what we would expect to get if we take a sample from the standard normal distribution) 1. 1 Course note Pr. Hadi, page

10 The three red points, observations no 62 (Malaysia)and no 43 (Guyana) on the right side and the observations no 50 (Ireland) are identified as outliers. They have a large residual, which is close or bigger than 3 in absolute value. The others residuals seem to be contained in a reasonable range. Another plot to analyse is the plot of residuals versus predicted values. This plot provides information about the validation of the assumptions of linearity and constant variances (homogeneity). The three same red points as above, are identified as outliers with again a large residual value Identification of high leverage points While large studentized residual indicates like analysed above, that the observation is an outlier, small residual do not necessarily indicate that the observation is not an outlier, in the contrary, a high leverage point have small residuals. A high-leverage point is an outlier in the X-space. This type of point can be identified by examining the leverage values. Let s analyse the plot of the leverage values versus the index. The 7 observations in green are high leverage points. The are from the left to the right the observations no 5 (Australia), no 40 (Greece), no 52 (Italy),no 78 (Paraguay), no 82 (Portugal), no 96 (Switzerland) and no 106 (USA). Theses observations should be flagged and then examined to see if they are also influential. Let s examine them by mean of a boxplot. 10

11 On the box plot, 6 points are revealed to be high leverage points, the first point on the top is observation no 82 (Portugal), then two observations together no 40 (Greece) and no 78 (Paraguay), then no 5 (Australia) and again two points together no 52 (Italy) and no 96 (Switzerland). Only no 106 (USA) does not appear on the box plot Identification of influential observations The three following figures represent the plots of the influence measures called Cook s distance, Dffits and Hadi s influence measure. 11

12 The outliers detected above are marked in red and the high leverage points in green. Theses two plots show the detection of 6 influential measures, which are no 78 (Paraguay), no 50 (Ireland), no 43 (Guyana), no 62 (Malaysia), no 82 (Portugal) and no 96 (Switzerland) Summary of the unusual observations The potential-residual plot allows to aid in classifying unusual observations as high-leverage points (green), outliers (red) or a combination of both. One can observe on the top left side of the chart the 7 high leverage points detected (green) and below on the right side the 3 outliers (red). Lets analyse now on the right graph below marked in orange the 6 influential values detected. The three outliers and three of the high leverages point have been identified as influencing measure (orange). As mentioned in the book Regression Analysis by Examples, Outliers should always be scrutinized carefully. Points with high leverage that are not influential do not cause problems. High leverage points that are influential should be investigated because these points are outlying as far as the predictor variables are concerned and also influence the fit. In order to get an idea of the sensitivity of the analysis to the points detected as influential, let s fit the model again without the 6 offending points. Table 9 shows details for the 6 observations detected as influential measures. I check the reporting of these data according to the internet site mentioned above and have not fund any reporting problem. The problem with these data depends maybe of the specificity of the countries. As I am not sure about the cause of the problem, I decide to drop the 6 influential measures and repeat the analysis. 12

13 3.5 Repeating the above steps without the 6 detected influential measures Model fit the the data Tables of the appendix show backward and stepwise selection models without the six influential measures. The model found does not change in terms of predictor chosen initially. An increase of the adjusted R 2 from 0,93 to 0,959 as well as of the t-test values has been observed. That means that the 6 influential measures dropped out allow to increase the quality of the model (which was already very good). Table 11 and 13 show also the absence of multi-collinearity in the model by mean of the VIF value, which is lower than 10 at the exception of ln_gdp (max. VIF value of 12,9). However ln_gdp has not been selected in the model, neither the first time nor now Diagnostics tests / plots Let s check again the assumptions of the model. Figures show the plots of residuals versus each of the predictors selected. On these plots one observe a quite random scatter of points, which are separated in 2 groups, on group for industrialised countries and the second group for less or non industrialised countries. Theses plot also shows the linearity of the two predictors. The variance of the variables looks homogenous and is stabilised (between 2,5 and 2,5 at the exception of one observation which is Chile). It is now important to check the normality assumption by examining the normal probability plot of the residuals and the histogram of the residuals on figures On these two plots one observes a nearly straight line and a normal distributed histogram, at the exception of one observation (Chile), which looks like an outlier but let s observe it again later. The assumption holds. If the regression model and the associated assumptions are valid, the points on the plot of studentized residuals versus predicted values should appear as a horizontal band showing no discernible pattern. Let s analyse this plot on figure 37. Once again everything looks good at the exception of one point, which is Chile again Identification of outliers, leverage points, influential measures As already mentioned above for the validation of the assumptions, everything looks good at the exception of one point, which is Chile. This point is marked in red in the two following figures and has been identified as an outlier: 13

14 Let s analyse now the plot of the leverage values versus the index in order to check the presence or not of high leverage point in this reduced data set: In this data set three points can be presented as high leverage points no 5 (Australia), no 40 (Greece) and no 106 (USA). These points are exactly the one already observed as high leverage points in the previous analysis but these points were not identified as influential measures. For this reasons, we let them in the model. Let s analyse now the three figures representing the plots for the identification of influential measures: Cook s distance, Dffits and Hadi s influence. 14

15 The three plots show principally two countries identified as influential measures Chile and Iceland. These points were not identified during the first analysis. In order to be sure about this last analysis, I suggest to look again at the potential-residual plot. The observations no 40 (Greece), no 106 (USA) are in the upper left side of the chart and identified as high leverage points and the observations no 21 (Chile), which is in the lower right side of the chart is an outlier. This analysis shows that a few points were again identified as influential measures. However the maximal leverage value is still very low and even if some observations influence the regression more than others, the magnitude of this influence has been strongly reduced by dropping 6 observations out of the model (ex: highest Cooks distance value before dropping the identified problematic observations was 0,225 and has been reduced to 0,1 and the same for Hadi s influence, which has been reduced from more than 0,4 to 0,3) Summary of the 2d fitting of the model without 6 influential measures However as the normality assumption showed a critical value, I suggest to retry the model selection without the observation no 21 (Chile), which as also been detected as an outliner and as an influential measure. 15

16 3.6 Repeating the above steps without the 7 detected influential measures Model fit the data I tried again a backward and a stepwise selection model and the result does not change. The two predictors chosen during the two previous steps stay the same. The good news is that the adjusted R 2 increases lightly again from 0,959 to 0,962 as well as the t-value for the sqr_cmr. Table 14 shows the results of the regression Diagnostics tests and plots, identification of outliers, high leverage points and influential measures Figures show the plots used for checking the validity of assumptions as well as critical obervations. All the assumptions for the validity of the least square regression model hold. The normality assumption and the homogeneity of the variance are now respected. No additional outlier has been discovered. The high influencing points are the same as for the second fitting of the model. The general magnitude of these influence has been strongly reduced with the 2 nd and 3 rd fitting by dropping out 7 influential measures. Figure 44 tells that there are still some points poorly fitted by the model, but the maximum squared residual is lower than before and the residuals are contained in a reasonable range. An alternative way to deal with critical observations without deleting them is to use robust methods (LMS, LAV, RIRLS).I tried to use the BACON / RIRLS algorithms and I got a very bad result (0 values and no outliers detected). I then decided to leave it. 4. Summary and conclusions 4.1 Final model The final model is the following and is the same as the one selected during the 1 st fitting of the data. This model has been built without 7 observations (detected as critical values), which have been dropped out in 2 steps. The 2 nd fitting has been done without 6 obervations and the 3 rd without 7 obervations (I dropped one more). (INTERNET) = 0,25 + 0,66* (COMPUTER) + 0,35* (MOBTEL) 4.2 Conclusions My fist hypothesis was that more than 3 variables were included in the final model. However the hypothesis that the variables COMPUTER and MOBTEL influence positively the INTERNET was correct. It is very 16

17 interesting to discover that from an economic point of view, lngdp or 1/CPI, were not chosen in the final model. That means that these variables influence poorly SQR_INTERNET. The adjusted R 2 is 96,2% after 3 fitting of data, which is a very good result. That means that the two selected variables explained 96,2% of the variation of SQR_INTERNET. The adjusted R 2 found after the 1 st fitting was already very good with 93%. It was very interesting to see that dropping out 7 critical values in 2 steps (fittings) could increase the quality of the model. It was also very interesting to oberve that after each of the 3 fittings, I always discovered critical values. The difficult decision is then to decide when to stop. A perfect model in statistic does not exist and it will never reflect the reality if we always decide to drop out variables. 17

18 5. Appendix 5.1 Annotated Computer outputs Tables Table 1: Summary of the initial variables Variable Count Mean StdDev Min Max INTERNET GDP POP e6 COMPUTER AREA e6 TELCOST CPI URBANPOP MOBTEL Table 2: Correlation matrix of the initial variables Table 3 : Summary of the transformed variables Table 4: Correlation matrix of the transformed variables 18

19 Table 5: Model summary of the backward selection method (1 st fitting) Model Summary Model Change Statistics Adjusted Std. Error of R Square R R Square R Square the Estimate Change F Change df1 df2 Sig. F Change,967 a,935,930 1,74854, , ,000,967 b,935,931 1,74089,000, ,764,967 c,935,932 1,73305,000, ,812,967 d,935,932 1,72631,000, ,675,966 e,934,932 1, ,001 2, ,147,966 f,932,930 1, ,002 2, ,108,965 g,931,930 1, ,001 2, ,126 a. Predictors: (Constant), URBPOP, LN_POP, NEW_CPI, TELCOST, SQR_MBL, LN_ARA, SQR_CMR, LNGDP b. Predictors: (Constant), URBPOP, NEW_CPI, TELCOST, SQR_MBL, LN_ARA, SQR_CMR, LNGDP c. Predictors: (Constant), URBPOP, NEW_CPI, TELCOST, SQR_MBL, SQR_CMR, LNGDP d. Predictors: (Constant), URBPOP, NEW_CPI, TELCOST, SQR_MBL, SQR_CMR e. Predictors: (Constant), URBPOP, TELCOST, SQR_MBL, SQR_CMR f. Predictors: (Constant), TELCOST, SQR_MBL, SQR_CMR g. Predictors: (Constant), SQR_MBL, SQR_CMR Table 6: Detailed analysis of the backward model selection steps (1 st fitting) 19

20 Model (Constant) LNGDP SQR_MBL LN_POP LN_ARA SQR_CMR NEW_CPI TELCOST URBPOP (Constant) LNGDP SQR_MBL LN_ARA SQR_CMR NEW_CPI TELCOST URBPOP (Constant) LNGDP SQR_MBL SQR_CMR NEW_CPI TELCOST URBPOP (Constant) SQR_MBL SQR_CMR NEW_CPI TELCOST URBPOP (Constant) SQR_MBL SQR_CMR TELCOST URBPOP (Constant) SQR_MBL SQR_CMR TELCOST (Constant) SQR_MBL SQR_CMR Unstandardized Coeff icients a. Dependent Variable: SQR_INT Standardi zed Coeff icien ts Coefficients a 95% Confidence Interv al f or B t Sig. Lower Bound Upper Bound Correlations Zero-order Partial Part Collinearity Statistics B Std. Error Beta Tolerance VIF -1,406 2,386 -,589,557-6,139 3,326,171,385,041,443,658 -,592,933,920,044,011,074 13,544,346,065,396 5,316,000,217,475,933,464,133,113 8,840-4,45E-02,148 -,011 -,302,764 -,337,248 -,080 -,030 -,008,441 2,266 4,926E-02,129,015,382,704 -,207,305 -,164,038,010,417 2,396,558,075,520 7,464,000,410,706,949,592,187,129 7, ,921 81,341,038 1,278,204-57, ,243,394,125,032,724 1,382-5,998 3,224 -,052-1,860,066-12,393,397,211 -,180 -,047,794 1,260 1,444E-02,013,049 1,120,265 -,011,040,683,110,028,329 3,040-1,562 2,320 -,673,502-6,162 3,039,172,383,041,449,654 -,587,931,920,044,011,074 13,542,343,064,392 5,370,000,216,469,933,466,134,117 8,565 2,127E-02,089,006,238,812 -,156,198 -,164,023,006,863 1,158,560,074,522 7,577,000,414,707,949,596,189,131 7, ,297 80,959,037 1,276,205-57, ,842,394,124,032,724 1,381-5,766 3,117 -,050-1,850,067-11,947,416,211 -,178 -,046,842 1,188 1,458E-02,013,049 1,136,259 -,011,040,683,111,028,329 3,036-1,288 2,005 -,642,522-5,264 2,689,159,377,038,421,675 -,589,906,920,041,010,075 13,253,344,063,393 5,437,000,219,470,933,469,135,118 8,497,559,073,520 7,619,000,413,704,949,597,189,132 7, ,812 80,355,037 1,267,208-57, ,141,394,123,031,729 1,372-5,723 3,098 -,050-1,847,068-11,866,420,211 -,177 -,046,844 1,184 1,539E-02,012,052 1,249,215 -,009,040,683,121,031,354 2,824 -,491,658 -,746,457-1,795,813,358,053,409 6,736,000,253,464,933,547,166,165 6,053,572,066,533 8,691,000,442,703,949,645,215,162 6, ,746 76,511,040 1,461,147-39, ,437,394,140,036,797 1,254-5,940 3,043 -,052-1,952,054-11,973,093,211 -,186 -,048,868 1,151 1,832E-02,010,062 1,813,073 -,002,038,683,173,045,522 1,916,175,476,368,713 -,769 1,120,356,053,407 6,659,000,250,462,933,541,165,165 6,047,595,064,554 9,257,000,468,722,949,667,230,172 5,810-5,054 2,998 -,044-1,686,095-10,997,889,211 -,161 -,042,904 1,106 1,630E-02,010,055 1,619,108 -,004,036,683,155,040,532 1,880,764,310 2,466,015,150 1,379,373,053,426 7,071,000,269,478,933,563,177,172 5,807,614,064,572 9,646,000,488,740,949,680,241,178 5,614-4,634 3,009 -,040-1,540,126-10,599 1,330,211 -,147 -,039,911 1,097,513,265 1,935,056 -,013 1,038,357,052,408 6,860,000,254,460,933,549,173,179 5,574,622,064,580 9,752,000,496,749,949,683,246,179 5,574 Table 7: Model summary of the stepwise selection method (1 st fitting) Model Summary Model 1 2 Adjusted Std. Error of R Square R R Square R Square the Estimate Change F Change df1 df2 Sig. F Change,949 a,901,900 2,09557, , ,000,965 b,931,930 1,75937,030 47, ,000 a. Predictors: (Constant), SQR_CMR b. Predictors: (Constant), SQR_CMR, SQR_MBL Change Statistics Table 8: Detailed analysis of the stepwise model selection steps (1 st fitting) 20

21 Model 1 2 (Constant) SQR_CMR (Constant) SQR_CMR SQR_MBL Unstandardized Coeff icients a. Dependent Variable: SQR_INT Standardi zed Coeff icien ts Coefficients a 95% Confidence Interv al f or B t Sig. Lower Bound Upper Bound Correlations Zero-order Partial Part Collinearity Statistics B Std. Error Beta Tolerance VIF,561,316 1,779,078 -,064 1,187 1,019,032,949 31,646,000,955 1,083,949,949,949 1,000 1,000,513,265 1,935,056 -,013 1,038,622,064,580 9,752,000,496,749,949,683,246,179 5,574,357,052,408 6,860,000,254,460,933,549,173,179 5,574 Table 9: Identifed influencial measures (1 st fitting) Multivariate Summaries No Selector 749 total cases of which 637 are missing Variable Count Sum Mean StdDev Min Max Skewness Kurtosis žint žcmr žmbl Table 10: Model summary of the backward selection method (2 nd fitting) Model Summary h Model Change Statistics Adjusted Std. Error of R Square R R Square R Square the Estimate Change F Change df1 df2 Sig. F Change,980 a,961,958 1,35658, , ,000,980 b,961,958 1,35034,000, ,752,980 c,961,959 1,34454,000, ,699,980 d,961,959 1,33930,000, ,638,980 e,960,959 1,33818,000, ,364,980 f,960,959 1,33521,000, ,461,980 g,959,959 1, ,001 1, ,161 a. Predictors: (Constant), URBPOP, LN_POP, NEW_CPI, TELCOST, SQR_MBL, LN_ARA, SQR_COMP, LNGDP b. Predictors: (Constant), URBPOP, LN_POP, NEW_CPI, TELCOST, SQR_MBL, SQR_COMP, LNGDP c. Predictors: (Constant), URBPOP, NEW_CPI, TELCOST, SQR_MBL, SQR_COMP, LNGDP d. Predictors: (Constant), URBPOP, NEW_CPI, TELCOST, SQR_MBL, SQR_COMP e. Predictors: (Constant), URBPOP, TELCOST, SQR_MBL, SQR_COMP f. Predictors: (Constant), URBPOP, SQR_MBL, SQR_COMP g. Predictors: (Constant), SQR_MBL, SQR_COMP h. Dependent Variable: SQR_INT 21

22 Table 11: Detailed analysis of the backward model selection steps (2 nd fitting) Model (Constant) LNGDP SQR_MBL LN_POP LN_ARA SQR_COMP NEW_CPI TELCOST URBPOP (Constant) LNGDP SQR_MBL LN_POP SQR_COMP NEW_CPI TELCOST URBPOP (Constant) LNGDP SQR_MBL SQR_COMP NEW_CPI TELCOST URBPOP (Constant) SQR_MBL SQR_COMP NEW_CPI TELCOST URBPOP (Constant) SQR_MBL SQR_COMP TELCOST URBPOP (Constant) SQR_MBL SQR_COMP URBPOP (Constant) SQR_MBL SQR_COMP a. Dependent Variable: SQR_INT Unstandardized Coeff icients Standardi zed Coeff icien ts Coefficients a 95% Confidence Interv al f or B t Sig. Lower Bound Upper Bound Correlations Zero-order Partial Part Collinearity Statistics B Std. Error Beta Tolerance VIF -1,357 1,879 -,722,472-5,086 2,372,148,302,035,489,626 -,451,746,927,050,010,077 12,939,336,057,379 5,941,000,224,449,952,517,119,099 10,114 5,922E-02,120,015,495,622 -,178,297 -,078,050,010,415 2,410-3,26E-02,103 -,010 -,317,752 -,237,172 -,153 -,032 -,006,399 2,506,608,064,558 9,464,000,481,736,967,693,190,116 8,642 44,650 63,735,016,701,485-81, ,146,372,071,014,727 1,375-1,669 2,592 -,015 -,644,521-6,814 3,476,203 -,065 -,013,776 1,288 1,014E-02,010,035,982,329 -,010,031,702,099,020,312 3,203-1,445 1,850 -,781,437-5,116 2,227,155,299,037,517,607 -,440,749,927,052,010,078 12,866,338,056,381 6,028,000,227,449,952,520,120,100 10,018 3,133E-02,081,008,388,699 -,129,192 -,078,039,008,903 1,107,607,064,557 9,504,000,480,734,967,693,190,116 8,623 46,436 63,193,017,735,464-78, ,841,372,074,015,733 1,364-1,868 2,504 -,016 -,746,457-6,837 3,100,203 -,075 -,015,825 1,213 9,489E-03,010,033,942,348 -,010,029,702,095,019,325 3,078-1,070 1,572 -,681,497-4,189 2,048,140,296,033,472,638 -,447,726,927,047,009,079 12,648,343,055,386 6,273,000,234,451,952,533,125,104 9,583,603,063,553 9,607,000,479,728,967,695,191,119 8,400 45,799 62,900,017,728,468-79, ,607,372,073,014,733 1,364-2,014 2,465 -,018 -,817,416-6,904 2,877,203 -,082 -,016,844 1,185 1,018E-02,010,035 1,031,305 -,009,030,702,103,020,335 2,982 -,369,514 -,719,474-1,388,649,356,047,401 7,576,000,262,449,952,604,150,140 7,135,614,058,563 10,533,000,498,730,967,725,209,137 7,296 54,568 59,861,020,912,364-64, ,330,372,091,018,803 1,245-2,222 2,416 -,020 -,920,360-7,014 2,571,203 -,092 -,018,871 1,148 1,284E-02,008,045 1,591,115 -,003,029,702,157,032,498 2,008-4,88E-02,374 -,130,896 -,790,693,353,047,398 7,544,000,260,446,952,600,149,141 7,114,627,057,575 11,083,000,515,739,967,741,219,145 6,877-1,745 2,356 -,015 -,741,461-6,420 2,929,203 -,073 -,015,914 1,094 1,181E-02,008,041 1,480,142 -,004,028,702,146,029,508 1,969 -,125,359 -,348,728 -,836,586,347,046,391 7,549,000,256,438,952,599,149,145 6,893,632,056,580 11,299,000,521,743,967,746,223,148 6,762 1,117E-02,008,039 1,410,161 -,005,027,702,138,028,514 1,946,289,207 1,399,165 -,121,699,361,045,407 7,999,000,272,451,952,619,159,152 6,577,645,055,592 11,637,000,535,755,967,754,231,152 6,577 Table 12: Model summary of the stepwise selection method (2 nd fitting) Model Summary c Model 1 2 Change Statistics Adjusted Std. Error of R Square R R Square R Square the Estimate Change F Change df1 df2 Sig. F Change,967 a,934,934 1,69996, , ,000,980 b,959,959 1,34161,025 63, ,000 a. Predictors: (Constant), SQR_COMP b. Predictors: (Constant), SQR_COMP, SQR_MBL c. Dependent Variable: SQR_INT Table 13: Detailed analysis of the stepwise model selection steps (2 nd fitting) 22

23 Model 1 2 (Constant) SQR_COMP (Constant) SQR_COMP SQR_MBL a. Dependent Variable: SQR_INT Unstandardized Coeff icients Standardi zed Coeff icien ts Coefficients a 95% Confidence Interv al f or B t Sig. Lower Bound Upper Bound Correlations Zero-order Partial Part Collinearity Statistics B Std. Error Beta Tolerance VIF,237,262,905,368 -,282,756 1,054,027,967 38,461,000,999 1,108,967,967,967 1,000 1,000,289,207 1,399,165 -,121,699,645,055,592 11,637,000,535,755,967,754,231,152 6,577,361,045,407 7,999,000,272,451,952,619,159,152 6,577 Table 14: Stepwise selection method for data set without 7 influencial measures (3 rd fitting) Model Summary c Model 1 2 Change Statistics Adjusted Std. Error of R Square R R Square R Square the Estimate Change F Change df1 df2 Sig. F Change,970 a,940,939 1,62505, , ,000,981 b,963,962 1,28194,023 63, ,000 a. Predictors: (Constant), SQR_COMP b. Predictors: (Constant), SQR_COMP, SQR_MBL c. Dependent Variable: SQR_INT Model 1 2 (Constant) SQR_COMP (Constant) SQR_COMP SQR_MBL Unstandardized Coeff icients a. Dependent Variable: SQR_INT Standardi zed Coeff icien ts Coefficients a 95% Confidence Interv al f or B t Sig. Lower Bound Upper Bound Correlations Zero-order Partial Part Collinearity Statistics B Std. Error Beta Tolerance VIF,194,251,775,440 -,303,692 1,052,026,970 40,191,000 1,001 1,104,970,970,970 1,000 1,000,253,198 1,279,204 -,139,646,662,053,610 12,434,000,556,767,970,776,237,151 6,635,346,043,391 7,970,000,260,432,952,619,152,151 6, Figures 23

24 Figure 1: Scatter plot INT vs GDP Figure 2: Scatter plot INT vs POP Figure 3: Scatter plot INT vs COMPUTER Figure 4: Scatter plot INT vs AREA 24

25 Figure 5: Scatter plot INT vs CPI Figure 6: Scatter plot INT vs URBANPOP Figure 7: Scatter plot INT vs TELCOST 25

26 Figure 8: Scatter plot INT vs MOBTEL Figure 9: Histogramm and normal probability plot of the transformed variable SQR_INT Figure 10: Histogramm and normal probability plot of the transformed variable Ln_GDP 26

27 Figure 11: Histogramm and normal probability plot of the transformed variable Ln_POP Figure 12: Histogramm and normal probability plot of the transformed variable SQR_CRM Figure 13: Histogramm and normal probability plot of the transformed variable Ln_ARA 27

28 Figure 14: Histogramm and normal probability plot of the transformed variable 1/CPI Figure 15: Histogramm and normal probability plot of the transformed variable SQR_MBL Figure 16: Histogramm and normal probability plot of the variable TELCOST 28

29 Figure 17: Histogramm and normal probability plot of the variable URBAN Figure 18: Scatterplot matrix of the transformed variables 29

30 SQR_INT LNGDP LN_POP SQR_MBL LN_ARA SQR_CMR NEW_CPI TELCOST URBPOP Figure 19: Added-variable plot Sqr(Int) residuals vs Ln_GDP residuals (1 st fitting) Figure 20: Added-variable plot Sqr(Int) residuals vs Ln_POP residuals (1 st fitting) Figure 21: Added-variable plot Sqr(Int) residuals vs SQR_CMR residuals (1 st fitting) 30

31 Figure 22: Added-variable plot Sqr(Int) residuals vs Ln_ARA residuals (1 st fitting) Figure 23: Added-variable plot Sqr(Int) residuals vs 1/ CPI residuals (1 st fitting) Figure 24: Added-variable plot Sqr(Int) residuals vs URP residuals (1 st fitting) Figure 25: Added-variable plot Sqr(Int) residuals vs SQR_MBL residuals (1 st fitting) 31

32 Figure 26: Added-variable plot Sqr(Int) residuals vs TELCOST residuals (1 st fitting) Figure 27: Scatter plot SQR_INT vs LnCMR (1 st fitting) Figure 28: Scatter plot SQR_INT vs SQR_MBL (1 st fitting) Figure 29: Plots of studentized residuals versus SQR_CMR and versus SQR_MBL (1 st fitting) 32

33 Figure 30: Plots of studentized residuals versus SQR_MBL (1 st fitting) Figure 31: Histogramm of the residuals (1 st fitting) 33

34 Figure 32: Scatter plot of the studentized residuals vs the normal scores (1 st fitting) Figure 33: Plolt of residuals vs. sqr_cmr (2 nd fitting) Figure 34: Plot of residuals vs sqr_mbl (2 nd fitting) Figure 35: Normal probability plot fo the residuals (2 nd fitting) 34

35 Figure 36: Histogramm of the residuals (2 nd fitting) Figure 37: Plot of residuals versus predicted values (2 nd fitting) Figure 38: Plot of residuals versus predictors (3 rd fitting) Figure 39: Normal probability plot and histogramm of the residuals (3 rd fitting) 35

36 Figure 40: Scatter plot of residuals vs predicted values (3 rd fitting) Figure 41: Plot of leverage values versus the index (3 rd fitting) Figure 42: Plots for identification of influcencial obervations (3 rd fitting) 36

37 Figure 43: Plot of Potential-Residuals (3 rd fitting) Figure 44: Leverage versus squared residual plot (3 rd fitting) 37

38 5.2 List of data set OBS_ NB CNTRY_NAME GDP POP COMPUTER AREA TELCOST CPI URBANPOP MOBTEL INTERNET 1 Albania Algeria Argentina Armenia Australia Austria Bangladesh Barbados Belgium Belize Bhutan Bolivia Botswana Brazil Bulgaria Burkina Faso Cambodia Cameroon Central African Republic Chad Chile Colombia Costa Rica Cote d'ivoire Croatia Cyprus Czech Republic Denmark Ecuador Egypt El Salvador Ethiopia Fiji Finland France Gabon Gambia Germany Ghana Greece Grenada Guatemala Guyana Honduras Hungary Iceland India Indonesia Iran (Islamic Republic of) Ireland

39 OBS_ NB CNTRY_NAME GDP POP COMPUTER AREA TELCOST CPI URBANPOP MOBTEL INTERNET 51 Israel Italy Jamaica Japan Jordan Kenya Lao People's Democratic Republic 58 Latvia Lithuania Luxembourg Malawi Malaysia Mauritania Mauritius Mexico Moldova, Republic of Mongolia Morocco Mozambique Namibia Nepal Netherlands New Zealand Nicaragua Norway Pakistan Panama Paraguay Peru Philippines Poland Portugal Romania Russian Federation Samoa Saudi Arabia Senegal Singapore Slovakia Slovenia Solomon Islands South Africa Spain Sri Lanka Sweden Switzerland Syrian Arab Republic Thailand Togo Trinidad and Tobago Tunisia Turkey

40 OBS_ NB CNTRY_NAME GDP POP COMPUTER AREA TELCOST CPI URBANPOP MOBTEL INTERNET 103 Uganda Ukraine United Kingdom of Great Britain and Northern Ireland 106 United States of America Uruguay Venezuela Viet Nam Yemen Zambia Zimbabwe