Correlation or Causality: New Evidence from Cross-Sectional Statistical Analysis of Auto-Crash Data in Nigeria

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1 Correlation or Causality: New Evidence from Cross-Sectional Statistical Analysis of Auto-Crash Data in Nigeria Olushina Olawale Awe, Department of Mathematics, ObafemiAwolowo University, Ile-Ife, Nigeria. Abstract Correlation Analysis is a key statistical technique for detecting the extent of relationship that exists among variables of interest in any environment and in various fields of human endeavor ranging from science, social sciences arts and even engineering and technology. This study is aimed at investigating the extent of association among some cross sectional, categorical, auto-crash variables in Nigeria. Differences between causation and correlation are re-examined. In all analyses, we find that all four variables considered are highly correlated. However, we note that correlation is not tantamount to causality. Finally, we compute the statistic of the coefficients of determination which is further used to determine the proportion of variation of one variable that is attributable to the variance in a related variable. Key words: Linear, Correlation, Classical Regression Models, Coefficient of Determination,Causality. 1. Introduction Controversy has continued to trail the exact number of people killed or injured due to fatal or serious accidents in Nigeria. The 2004 edition of World Health Day, organized by the World Health Organization (WHO), was for the first time devoted to issues on Road Safety.

2 Every year, according to statistics, about 1.2 million people are known to die in road accidents worldwide. Millions of others sustain severe injuries,while some others suffer permanent disabilities due to number of fatal and serious cases of road accident. Enormous human potential is being destroyed, which is also accompanied by grave social and economic consequences. Road safety is thus a major public health issue throughout the world. The incidence of death and injury as a result of road accident is now acknowledged to be a global phenomenon with authorities in virtually all countries of the world concerned about the growth in the number of people killed and seriously injured on their roads. In recent years there has been two major studies of causes of death worldwide which has been covered in thein the World Health report Making a Difference (WHO 1999). These publications show that in 1990, road accidents as cause of death or disability were by no means insignificant, lying in ninth place out over 100 separately identified causes. However, by the year 2020 forecasts suggest that as a cause of death, road accidents will move to sixth place. In this paper, we investigate the dynamic interconnectedness between some road accident data. The data is classified into four different categories as recorded by the Federal Road Safety Corporation of Nigeria (FRSCN) and the Nigerian police viz. Fatal, Killed, Injured and Serious. The rest of the paper is organized as follows: Section two discusses the preliminary concept of correlation analysis, causality and coefficient of determination, section three is on the data used in the study, section four discusses the findings in this study while section five concludes. 2.0 Concept of Correlation Analysis. Correlation analysis, as depicted in literature, is a very important tool in statistics for studying the relationship between two or more variables.it is generally classified into two types:bivariate and Partial Correlation.Bivariate correlation shows an association between two variables while partial correlation shows the association between two variables while keeping control or adjusting the effect of one or more additional variables. A correlation is a degree of measure. This means it can be positive, negative or perfect. In a positive

3 correlation, if there is an increase(or decrease) in one variable, then there is a simultaneous increase or decrease in the other variable.in a negative correlation, if there is a decrease in one variable, there is a corresponding increase in the other. A perfect correlation is that type of correlation where a change in one variable affects an equivalent change in another. The value of the correlation coefficient does not depend on the change in origin and the change in scale. A correlation problem arises when a researcher begins to ask himself whether there is any relationship between a pair of variables that interests him. For instance, we may be interested in asking:is there any relationship between smoking, alcohol intake and academic performance of students or incidence of minor andfatal road accidents cases? The investigation of the relationship between two variables such as these usually begins with an attempt to discover the approximation form of the relationship by graphing the data as points in the X,Y plane. This can help us to discern whether there is any pronounced relationship and if so, whether the relationship may be treated as approximately linear. The strength of the relationship is given by the magnitude of r(correlation coefficient). The sign of rmerely tells us whether the values of a variable y tend to increase or decrease withthe values of another variable, x. 2.1Concept of Causality: Difference between Causality and Correlation Causality is the relationship between cause and effect. One of the most common errors often found in many scientific and environmental studies is the confusion between correlation and causation. In theory, these are easy to distinguish - an action or occurrence can cause another -such as smoking causes lung cancer, or it can correlate with another -such as smoking is correlated with alcoholism. If one action causes another, then they are most certainly correlated. But just because two things occur together does not mean that one caused the other, even if the correlation is very high. Many studies are actually designed to test correlations. It is commonplace and obvious that correlation does not imply causality. In general, it is extremely difficult to establish causality between two correlated events or observances, although there are many statistical tools to establish a statistical significant correlation.there are several reasons why common sense conclusions about cause and effect might be wrong. Correlated occurrences may be

4 due to a common cause. For example, the fact that red hair is correlated with blue eyes stems from a common genetic specification which suffices for both. A correlation may also be observed when there is causality behind it for instance, it has been well established that cigarette smoking not only correlates with lung cancer, but actually causes it. But in order to establish cause, we would have to rule out the possibility that smokers are more likely to live in urban areas, where there is more pollution or any other possible explanation for the observed correlation. More so,it is quite possible for two variables to have zero correlation and yet for one of them to be completely determined by the other. A classic unresolved issue in structural equation modeling is the extent to which structural models can warrant causal claims(hayduk et al, 2003).In many real life cases, it seems obvious that one action causes another andadditional research would be needed to make a convincing argument for causality against mere correlation. Specifying a trustworthy basis on which researchers can move from correlational data to causal claims has remained elusive in recent times. Merely asserting a presupposed causal direction, as in the classical regression models is clearly insufficient to support claims of causal effects. In many instances, people are much more likely to jump to causal conclusions. The media often concludes a causal relationship among correlated observations when causality was not even considered by the study itself. Without clear reasons to accept causality, we should only accept correlation. Two events occurring in close proximity, like we have in this study, does not imply that one caused the other, even if it seems possible and sensible. Many articles have been written on causality and causal relations in various fields. They include: Granger CWJ(1969), Granger CWJ(1980), Granger CWJ( 1988), Swanson and Granger(1997), Entner et al (2010), Mohammed et al(2010), Chu and Chymour (2008), Arnold et al (2007), Eichler and Didelez(2009), Clarke and Mirza (2006), Erdal et al (2008), Pearl(2012) just to mention a few. Others include: Shojaie and Michailidis (2010), Moneta et al (2011),Chen and Hsiao (2010),White et al(2011), Zou et al (2010), Havackova- Schindler et al (2007), Haufe et al (2010),Eichler and Didelez (2007),Cheng(1996),Cheng et al(1997),toda et al (1994) etc.

5 Hinkelmann and Kempthorne (2008) highlighted nine basic definitions of causality as follows: A relation between events, process or entities in the same time series, subject to several conditions. A relationship between events, processes or entities in a time series such that when one occurs, the other follows invariably. A relationship among variables such that one has the efficacy to produce or alter another. A relationship among variables such that without one, the other could not occur. A relationship between experienced events, processes or entities and extraexperiential events, processes or entities. A relation between something and itself (self-causality). A relation between an event, process or entity and the reason or explanation for it. A relation between an idea and an experience. A principle or category incorporating into experience one of the previous ones Methodology: Estimation ofthe OptimalCorrelation Coefficient. If a linear relationship is assumed between two variables, the correlation coefficient can be computed by the formula:

6 r = χχχχ ( χχ 2 )( yy 2 ).. (1) whereχχ = X-XX and y = Y-YY ifss XXXX = XXXX NN, SS XX= XX2 NN and SS YY= YY2 NN.(2) thenss XX and SS YY is recognized as the standard deviations of the variables X and Y respectively while SS XX 2 aaaaaass YY 2 are their variances. The new quantity SS XXXX is called the covariance of X and Y.(see Spiegel and Stephen, 1998, p 315).Hence, inputting (2),(1) can be written as: r = SS XXXX SS XX SS YY.. (3) A short computational formula for computing rcan also be given by: r = NN XXYY ( XX)( YY) NN XX 2 ( XX) 2 [NN YY 2 ( YY) 2 ]. (4) If the value of the covariance of two variables is zero then the variables have no correlation. For the case of linear correlation, the quantity r is the same regardless of whether Y or X is considered independent or dependent variable. Therefore, r is a good measure of the linear relationship between two variables (Spiegel and Stephen, 1998). There are certain assumptions that come along with the correlation coefficient. The correlation coefficient assumes that the variables under study should be linearly related. A cause and effect relationship is assumed to exist between different forces operating on the variables. If there is no correlation between the variables, then the change in one variable will not affect the change in the other variable and therefore, the variables will be independent. We wish to point out in this work that a high correlation coefficient (near 1) does not necessarily indicate a direct dependence of the variables. Therefore, the high correlation of between number of fatal cases and number of people killed and the high coefficient (0.943) between injured and killed does not mean direct dependence or casualty. Correlation

7 merely shows the extent of relationship among variables but does not depict causation.vernoy and Kyle (2002) Coefficient of Determination(rr 22 ) The coefficient of determination is the square of the correlation coefficient. Much as correlation can be used to predict associations, they cannot be used to explain causation. It would be preposterous to assert that since one variable is highly correlated to another, then it causes it. Correlation merely establishes relationship and not causation. It is possible to measure the accuracy of associationbetween two variables by using the coefficient of determination to determine the part of variance of one variable that is attributable to the variance of a related variable, as we did here in this work. Coefficient of determination, also known as goodness of fit, helps to measure how well the sample regression line fits the data. rr 2 showsthe proportion of variation in Y that is explained by X. when rr 2 = 1 then the sample regression line fits the data perfectly, and if rr 2 = 0, then there is no overlap between Y and X. rr 2 lies between 0 and 1 i.e. 0 rr 2 1. rr 2 = nn (YY ii=1 ii YY ) 2 = RRRRRR.(5) nn ii=1(yy ii YY ) 2 TTTTTT Consider YY ii = ββ 0+ββ 1XX ii (6) Ifrr 2 = 1, then YY ii = YY ii. On the other hand, ifrr 2 = 0, it means there is no relationship between Y and X. rr 2 is the square of simple correlation coefficient. i.e.rr = ± rr 2 3.Data and Descriptive Statistics. The data used in this study is a cross-sectional data involving four attributes of auto-crash cases across Thirty Six States of Nigeria. It is a categorical secondary data obtained from the Federal Road Safety Corporation of Nigeria(FRSCN) and the Nigerian Police Force, Headquarters,Abuja.The degrees of Auto-Crash results across Thirty Six States of Nigeria are classified into four categories as follows: Fatal, Serious, Injured and Killed.There was no

8 data recorded for Taraba State. We have about 144 data points arranged and categorized into four different columns (see Appendix 2) 4. Results and Findings. Our findings reveal that all the four variables considered in this study are highly correlated. With the assumption that the variables are normally distributed, we computed both Pearson s Product Moment Correlation and Spearman s Rho.From the table of the correlation matrix of Pearson s Product Moment Correlations, we find that the bivariate correlation coefficient between the variables: fatal and serious is r fs = This indicates that there is a high co-movement between the two variables. As the number of fatal road accident increases in Nigeria, so also does the number of serious accidents. Their coefficient of determination R 2 =0.74.This index indicates that 7 of the variance in the number of fatal accidents can be explained by the variance in number of serious road accidents in Nigeria. The remaining 2 can be attributed to other factors. The bivariate correlation coefficient between the variables: fatal and injured is r fi = This indicates that there is a very high co-movement between the two variables. It implies that as the number of fatal road accident increases in Nigeria, so also does the increase in number of injured persons. Their coefficient of determination R 2 =0.86.This index indicates that 86% of the variance in the number of fatal accidents can be explained by the variance in number of persons injured in road accidents in Nigeria. The remaining 1 can be attributed to other factors. The bivariate correlation coefficient between the variables: fatal and killed isr fk = This indicates that there is high positive co-movement between the two variables. It implies that as the number of fatal road accident increases in Nigeria, so also does the number of people killed by road accidents. Their coefficient of determination R 2 =0.81.This index indicates that 8 of the variance in the number of fatal accidents can be explained by the variance

9 in number of persons killed due to road accidents in Nigeria. The remaining 19% can be attributed to other natural factors. The bivariate correlation coefficient between the variables: serious and injured is r si = This indicates that there is a high positive correlation between the two variables. It implies that as the number of serious road accident increases in Nigeria, so also does the number of injured persons due to road accidents. Their coefficient of determinationr 2 =0.71.This indicates that 7 of the variance in the number of serious accidents can be explained by the variance in number of persons injured due to road accidents in Nigeria. The remaining 29% can be attributed to other stochastic factors. The bivariate correlation coefficient between the variables: serious and killed is r sk =0.723.This figure indicates that there is a high positive correlation between the two variables. It implies that as the number of serious road accident increases in Nigeria, so also does the number of killed persons due to road accidents in Nigeria. Their coefficient of determination R 2 =0.52.This indicates that 5 of the variance in the number of serious accidents can be explained by the variance in number of persons killed due to road accidents in Nigeria. The remaining 58% can be attributed to other factors. Also, the bivariate correlation coefficient between the variables: injured and killed is r ik =0.943.This indicates that there is a high positive correlation between the two variables. It implies that as the number of people injured in road accident increase in Nigeria, so also does the number of people killed due to road accidents. Their coefficient of determination R 2 =0.889.This indicates that 89% of the variance in the number injured due to auto-crash can be explained by the variance in number of persons killed due to road accidents in Nigeria. The remaining 1 can be attributed to other factors. 5.Conclusion. Many revealing results have been unfolded in this work. We have deduced that there is high association among the incidence of some categorical auto-crash variables in Nigeria. A decrease in one leads to a decrease in the other, while a corresponding increase in one leads to a simultaneous increase in the other. If there is a reduction in the number of fatal

10 accidents in Nigeria for instance, it will lead to a corresponding decrease in the number of people killed in road accidents since both variables, fatal and killed are highly positively correlated. (see correlation matrix in appendix 1 below r=0.898 ).They move in the same direction. Same condition applies to serious and injured also. However, it should benoted that the interpretation of a correlation coefficient as a measure of the strength of the linear relationship between two or more variables is purely a mathematical interpretation and is completely devoid of any cause or effect implications.the fact that two variables tend to increase or decrease together does not imply that one has any direct or indirect effect on the other. Both may be influenced by other variables in such a manner as to give rise to a string of mathematical relationships. The correlation coefficient is merely concerned with determining how strongly two or more variables are linearly related but not capable of solving prediction problems which is the core of causality analysis. References Aalen OO (1987) Dynamic modeling and causality.scandactuarj : Aalen OO, Frigessi A (2007) What can statistics contribute to a causal understanding. Scand J Stat.34: Bartz, A. E (1989).Basic Statistical Concepts 2nd edition, Burgers Publishing Company, Minneapolis, USA. Ben Vogelvang, Econometrics Theory and Applications. Pearson Education Limited. Box GEP, Jenkins GM, Reinsel GC (1994) Time Series Analysis, Forecasting and Control. 3. Prentice Hall, Bruce E. Hansen (2000), Econometrics. University of Wisconsin Press. Cambridge, MA. With additional material by David Heckerman, Christopher Meek,

11 Gregory F Cooper Chris Brooks (2008), Introductory Econometrics for Finance (2nd Edition). Cambridge University Press. Commenges D, Gegout-Petit A (2007) A general dynamical statistical model with possible causal interpretation Comte F, Renault E (1996) Noncausality in continuous time models.econometr Theory 12: Cox DR, Wermuth N (1996) Multivariate dependencies models analysis and interpretation. Chapman & Dawid AP (1979) Conditional independence in statistical theory (with discussion). J R Stat Soc B 41:1 31 Dawid AP (2000) Causal Inference without Counterfactuals. J Am Stat Assoc 86:9 26 Dawid AP (2002) Influence diagrams for causal modelling and inference. Int Stat Rev 70: Dawid AP and Didelez V (2005) Identifying the consequences of dynamic treatment strategies, Technical Didelez V (2007) Graphical models for composable finite markov processes. Scand J Stat 34: Didelez V (2008) Graphical models for marked point processes based on local independence. J R Stat Soc. 70: Econometrica 37: Egbo, I and Egbo M (2010).Probability and Statistical Inference for Engineering and the Sciences. Milestones Publications limited, Oweri, Nigeria. Eichler M (2001) Graphical Modelling of Multivariate Time Series, Technical report. Universität Heidelberg Eichler M (2006) Graphical modelling of dynamic relationships in multivariate time series. In: Winterhalder M, Schelter B, Timmer J (eds) Handbook of time series analysis. Wiley- VCH, pp Eichler M (2007) Granger causality and path diagrams for multivariate time series. J Econometr 137:

12 Eichler M (2009) Causal inference from multivariate time series: what can be learned from granger causality. In: Glymour C, Wang W, Westerståhl D (eds) Proceedings from the 13th international congress of logic, methodology and philosophy of science. King s College Publications, London Eichler M, Didelez V (2007) Causal reasoning in graphical time series models. Proceedings of the 23 rd conference on uncertainty in artificial intelligence Englewood Cliffs Fisher R. A. (1925) Statistical Methods for Research workers, Oliver and Boyd, Edinburgh. Florens JP, Fougère D. (1996) Noncausality in continuous time. Econometrica 64: Florens JP, Mouchart M (1982) A note on noncausality. Econometrica 50: Florens JP, Mouchart M, Rolin JM (1990) Elements of Bayesian statistics. Marcel Dekker, New York Granger CWJ (1969) Investigating causal relations by econometric models and crossspectral methods. Granger CWJ (1980) Testing for causality, a personal viewpoint. J Econ Dyn Control 2: Granger CWJ (1986) Comment on Statistics and causal inference by P. Holland. J Am Stat Assoc 81: Granger CWJ (1988) Some recent developments in a concept of causality. J Econometr 39: Gujarati D. N.(1995),Basic Econometrics, McGraw-Hill, Inc. Hall, S. (2011). What Statistical Tools of Analysis are used in Survey Research? Downloaded from Heckman JJ (2008) Econometric causality.int Stat Rev 76:1 27 Heckman JJ, Navarro S (2007) Dynamic discrete choice and dynamic treatments effect. J Econon 136:

13 Hernán MA, Hernández-Díaz S, Robins JM (2004) A structural approach to selection bias. Epidemiology 15: Hernán MA, Taubman SL (2008) Does obesity shorten life? the importance of well defined interventions to answer causal questions. Int J Obes 32:8 14 Hsiao C (1982) Autoregressive modeling and causal ordering of econometric variables. J Econ Dyn Control 4: Kallenberg O (2001) Foundations of modern probability.2nd edn. Springer, New York Lauritzen SL (1996) Graphical models. Oxford University Press, Oxford Lauritzen SL (2001) Causal inference from graphical models. In: Barndorff-Nielsen OE, Cox DR, Klüppelberg C (eds) Complex stochastic systems. CRC Press, London pp Pearl J (1993) Graphical models, causality and interventions. Stat Sci 8: Pearl J (1995) Causal diagrams for empirical research (with discussion). Biometrika 82: Pearl J (2000) Causality. Cambridge University Press, Cambridge, UK Richardson T (2003) Markov properties for acyclic directed mixed graphs.scand J Stat 30: Robins JM (1986) A new approach to causal inference in mortality studies with sustained exposure periods application to control for the healthy worker survivor effect. Math Model 7: Robins JM, Hernán MA, Siebert U (2004) Effects of multiple interventions. In: Ezzati M, Lopez AD, Rubin DB (1974) Estimating causal effects of treatments in randomized and nonrandomized studies. J Edu Rubin DB (1978) Bayesian inference for causal effects: the role of randomization. Ann Stat 6:34 68 Schweder T (1970) Composablemarkov processes. J ApplProb 7: Sherman, R & Webb, R (1988) Qualitative Research in Education Forms and Methods. Lewes Falwes Press.

14 Spiegel, M.R. and Stephens, L.J.Schaum s Outlines of Theory and Problems of Statistics.3 rd edition.mcgraw-hill. Spirtes P, Glymour C, Scheines R (2001) Causation, prediction, and search, 2nd edn. [MIT Press, Vernoy, M & Kyle, D. J (2002).Behavioural Statistics in Action 3rd Edition. Boston: McGraw-Hill Higher Education. White H (2006): Time series estimation of the effects of natural experiments. J Econometrics 135: Williams H. Greene, Econometric Analysis. New York University Press. Appendix 1 Table 1:Matrix of Pearson Correlation coefficients Fatal Serious Injured Killed Fatal Pearson Correlation **.928 **.898 ** Sig. (2-tailed) N Serious Pearson Correlation.862 ** **.723 ** Sig. (2-tailed) N Injured Pearson Correlation.928 **.840 ** ** Sig. (2-tailed) N Killed Pearson Correlation.898 **.723 **.943 ** 1

15 Sig. (2-tailed) N **. Correlation is significant at the 0.01 level (2-tailed). Table 2:Matrix of Spearman s Correlations Fatal Serious Injured Killed Spearman's rho Fatal Correlation Coefficient **.938 **.949 ** Sig. (2-tailed) N Serious Correlation Coefficient.853 ** **.804 ** Sig. (2-tailed) N Injured Correlation Coefficient.938 **.855 ** ** Sig. (2-tailed) N

16 Killed Correlation Coefficient.949 **.804 **.914 ** Sig. (2-tailed) N **. Correlation is significant at the 0.01 level (2-tailed). Appendix 2 Table 3: Data on Recent Auto-Crash Statistics in Nigeria. State Fatal Injured Killed

17 Serious Abuja Abia Adamawa AkwaIbom Anambra Bauchi Bayelsa Benue Borno Cross River Delta Ebonyi Edo Ekiti Enugu Gombe Imo Jigawa Kaduna Kano Katsina Kebbi Kogi Kwara Lagos Nassarawa Niger Ogun Ondo

18 Osun Oyo Plateau Rivers Sokoto Yobe Zamfara Fatal Serious Killed Injured Abuja Adamawa Anambra Bayelsa Borno Delta Edo Enugu Imo Kaduna Katsina Kogi Lagos Niger Ondo Oyo Rivers Yobe Figure 1: Showing the incidence rate of recent accident statistics in some states in Nigeria. Lagos State has the highest incident rate.

19 Osun Ogun 9% Niger Nassarawa 6% Kwara Oyo Rivers Plateau Ondo Lagos 7% Kogi Kebbi 0% Yobe Sokoto Killed Zamfara Katsina Kano 7% Abuja Abia KadunaJigawa Adamawa 7% Akwa Ibom Imo Anambra Bauchi Bayelsa 0% Benue Borno Cross River Delta Ebonyi Ekiti 0% Gombe Enugu Edo Figure 2: Pie Chart showing percentage of people killed in road accident in Nigeria. It can be seen that Ogun State has the highest percentage followed by Lagos, Kano, Adamawa and Nassarawa. Ogun 9% Niger Nassarawa 5% Kwara Kogi Plateau Rivers Osun Ondo Kebbi 0% Lagos 9% Oyo Sokoto Yobe Katsina Jigawa Injured Zamfara Kano 8% Kaduna Abuja Abia Imo Adamawa Akwa 6% Ibom Anambra Enugu Gombe Bauchi Bayelsa 0% Benue 6% Cross Borno River Delta Ebonyi Edo Ekiti Figure 3: Pie Chart showing the percentage of people injured as a result of recent road accidents in Nigeria.

20 Lagos State has the highest percentage rate, followed by Kano State in the figure above. Rivers Niger Nassarawa 0% Sokoto Osun Plateau Ogun 7% Ondo Kwara Kogi Lagos 17% Oyo Kebbi 0% Yobe Serious Zamfara 0% Katsina Kano 8% Abia Kaduna Abuja Adamawa Akwa Ibom Jigawa Anambra Bauchi Enugu Imo Benue Bayelsa 6% 0% Borno Delta 6% Cross River Ebonyi Edo 6% Ekiti Gombe Figure 4: Pie Chart showing the percentage of serious accidents in Nigeria. Lagos State has the highest rate followed by Kano and Ogun State. Ogun 6% Niger Nassarawa 5% Ondo Osun Rivers Sokoto Oyo Lagos 8% Plateau Kwara KogiKaduna Kebbi 0% Katsina Fatal Yobe Zamfara Kano 9% Abuja Abia Akwa Ibom Jigawa Imo Adamawa Gombe Anambra Bauchi Bayelsa 0% Edo Cross River Delta 6% Enugu Benue 5% Borno Ebonyi Ekiti Figure 5: Pie Chart showing the percentage of Fatal Accidents in Nigeria. We notice that Kano State has the highest volume of fatal accident closely followed by Lagos and Delta.

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