496 STATISTICAL ANALYSIS OF CAUSE AND EFFECT

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1 496 STATISTICAL ANALYSIS OF CAUSE AND EFFECT * Use a non-parametric technique. There are statistical methods, called non-parametric methods, that don t make any assumptions about the underlying distribution of the data. Rather than evaluating the differences of parameters such as the mean or variance, non-parametric methods use other comparisons. For example, if the observations are paired they may be compared directly to see if the after is di erent than the before. Or the method might examine the pattern of points above and below the median to see if the before and after values are randomly scattered in the two regions. Or ranks might be analyzed. Non-parametric statistical methods are discussed later in this chapter. Equal variance assumption Many statistical techniques assume equal variances. ANOVA tests the hypothesis that the means are equal, not that variances are equal. In addition to assuming normality, ANOVA assumes that variances are equal for each treatment. Models fitted by regression analysis are evaluated partly by looking for equal variances of residuals for different levels of Xs and Y. Minitab s test for equal variances is found in Stat > ANOVA > Test for Equal Variances. You need a column containing the data and one or more columns specifying the factor level for each data point. If the data have already passed the normality test, use the P-value from Bartlett s test to test the equal variances assumption. Otherwise, use the P-value from Levene s test. The test shown in Figure 14.3 involved five factor levels and Minitab shows a confidence interval bar for sigma of each of the five samples; the tick mark in the center of the bar represents the sample sigma. These are the data from the sample of 100 analyzed earlier and found to be normally distributed, so Bartlett s test can be used. The P-value from Bartlett s test is 0.182, indicating that we can expect this much variability from populations with equal variances 18.2% of the time. Since this is greater than 5%, we fail to reject the null hypothesis of equal variances. Had the data not been normally distributed we would ve used Levene s test, which has a P-value of and leads to the same conclusion. REGRESSION AND CORRELATION ANALYSIS Scatter plots DefinitionöA scatter diagram is a plot of one variable versus another. One variable is called the independent variable and it is usually shown on the horizontal (bottom) axis. The other variable is called the dependent variable and it is shown on the vertical (side) axis.

2 Regression and correlation analysis 497 Figure Output from Minitab s test for equal variances. UsageöScatter diagrams are used to evaluate cause and effect relationships. The assumption is that the independent variable is causing a change in the dependent variable. Scatter plots are used to answer such questions as Does vendor A s material machine better than vendor B s? Does the length of training have anything to do with the amount of scrap an operator makes? and so on. HOW TO CONSTRUCT A SCATTER DIAGRAM 1. Gather several paired sets of observations, preferably 20 or more. A paired set is one where the dependent variable can be directly tied to the independent variable. 2. Find the largest and smallest independent variable and the largest and smallest dependent variable. 3. Construct the vertical and horizontal axes so that the smallest and largest values can be plotted. Figure 14.4 shows the basic structure of a scatter diagram. 4. Plot the data by placing a mark at the point corresponding to each X^Y pair, as illustrated by Figure If more than one classi cation is used, you may use di erent symbols to represent each group.

3 498 STATISTICAL ANALYSIS OF CAUSE AND EFFECT Figure Layout of a scatter diagram. Figure Plotting points on a scatter diagram. From Pyzdek s Guide to SPCöVolume One: Fundamentals,p.66. Copyright # 1990 by Thomas Pyzdek.

4 Regression and correlation analysis 499 EXAMPLE OF A SCATTER DIAGRAM The orchard manager has been keeping track of the weight of peaches on a day by day basis. The data are provided in Table Table Raw data for scatter diagram. From Pyzdek s Guide to SPCöVolume One: Fundamentals,p.67. Copyright # 1990 by Thomas Pyzdek. NUMBER DAYS ON TREE WEIGHT (OUNCES) Organize the data into X^Y pairs, as shown in Table The independent variable, X, is the number of days the fruit has been on the tree. The dependent variable, Y, is the weight of the peach. 2. Find the largest and smallest values for each data set. The largest and smallest values from Table 14.1 are shown in Table 14.2.

5 500 STATISTICAL ANALYSIS OF CAUSE AND EFFECT Table Smallest and largest values. From Pyzdek s Guide to SPCöVolume One: Fundamentals,p.68. Copyright # 1990 by Thomas Pyzdek. VARIABLE SMALLEST LARGEST Days on tree (X) Weight of peach (Y) Construct the axes. In this case, we need a horizontal axis that allows us to cover the range from 75 to 90 days. The vertical axis must cover the smallest of the small weights (4.4 ounces) to the largest of the weights (6.1 ounces). We will select values beyond these minimum requirements, because we want to estimate how long it will take for a peach to reach 6.5 ounces. 4. Plot the data. The completed scatter diagram is shown in Figure POINTERS FOR USING SCATTER DIAGRAMS. Scatter diagrams display di erent patterns that must be interpreted; Figure 14.7 provides a scatter diagram interpretation guide. Figure Completed scatter diagram. From Pyzdek s Guide to SPCöVolume One: Fundamentals,p.68. Copyright # 1990 by Thomas Pyzdek.

6 Regression and correlation analysis 501 Figure Scatter diagram interpretation guide. From Pyzdek s Guide to SPCöVolume One: Fundamentals,p.69. Copyright # 1990 by Thomas Pyzdek.. Be sure that the independent variable, X, is varied over a su ciently large range. When X is changed only a small amount, you may not see a correlation with Y, even though the correlation really does exist.. If you make a prediction for Y, for an X value that lies outside of the range you tested, be advised that the prediction is highly questionable and should be tested thoroughly. Predicting a Y value beyond the X range actually tested is called extrapolation.. Keep an eye out for the e ect of variables you didn t evaluate. Often, an uncontrolled variable will wipe out the e ect of your X variable. It is also possible that an uncontrolled variable will be causing the e ect and you will mistake the X variable you are controlling as the true cause. This problem is much less likely to occur if you choose X levels at random. An example of this is our peaches. It is possible that any number of variables changed steadily over the time period investigated. It is possible that these variables, and not the independent variable, are responsible for the weight gain (e.g., was fertilizer added periodically during the time period investigated?).

7 502 STATISTICAL ANALYSIS OF CAUSE AND EFFECT. Beware of happenstance data! Happenstance data are data that were collectedinthepastforapurposedi erentthanconstructingascatterdiagram. Since little or no control was exercised over important variables, you may nd nearly anything. Happenstance data should be used only to get ideas for further investigation, never for reaching nal conclusions. One common problem with happenstance data is that the variable that is truly important is not recorded. For example, records might show a correlation between the defect rate and the shift. However, perhaps the real cause of defects is theambienttemperature, whichalsochanges withtheshift.. If there is more than one possible source for the dependent variable, try using di erent plotting symbols for each source. For example, if the orchard manager knew that some peaches were taken from trees near a busy highway, he could use a di erent symbol for those peaches. He might nd an interaction, that is, perhaps the peaches from trees near the highway have a di erent growth rate than those from trees deep within the orchard. Although it is possible to do advanced analysis without plotting the scatter diagram, this is generally bad practice. This misses the enormous learning opportunity provided by the graphical analysis of the data. Correlation and regression Correlation analysis (the study of the strength of the linear relationships among variables) and regression analysis (modeling the relationship between one or more independent variables and a dependent variable) are activities of considerable importance in Six Sigma. A regression problem considers the frequency distributions of one variable when another is held fixed at each of several levels. A correlation problem considers the joint variation of two variables, neither of which is restricted by the experimenter. Correlation and regression analyses are designed to assist the analyst in studying cause and effect. They may be employed in all stages of the problem-solving and planning process. Of course, statistics cannot by themselves establish cause and effect. Proving cause and effect requires sound scientific understanding of the situation at hand. The statistical methods described in this section assist the analyst in performing this task. LINEAR MODELS A linear model is simply an expression of a type of association between two variables, x and y.alinear relationship simply means that a change of a given size in x produces a proportionate change in y. Linear models have the form:

8 512 STATISTICAL ANALYSIS OF CAUSE AND EFFECT ANOVA, or ANalysis Of VArianceöa table examining the hypothesis that the variation explained by the regression is zero. If this is so, then the observed association could be explained by chance alone. The rows and columns are those of a standard one-factor ANOVA table (see Chapter 17). For this example, the important item is the column labeled Significance F. The value shown, 0.00, indicates that the probability of getting these results due to chance alone is less than 0.01; i.e., the association is probably not due to chance alone. Note that the ANOVA applies to the entire model, not to the individual variables. The next table in the output examines each of the terms in the linear model separately. The intercept is as described above, and corresponds to our term a in the linear equation. Our model uses two independent variables. In our terminology staff ¼ b 1, food ¼ b 2. Thus, reading from the coefficients column, the linear model is: y ¼ 1:188 þ 0:902 staff score food score. The remaining columns test the hypotheses that each coefficient in the model is actually zero. Standard error columnögives the standard deviations of each term, i.e., the standard deviation of the intercept ¼ 0.565, etc. tstatcolumnöthe coefficient divided by the standard error, i.e., it shows how many standard deviations the observed coefficient is from zero. P-valueöshowstheareainthetailofat distribution beyond the computed t value. For most experimental work, a P-value less than 0.05 is accepted as an indication that the coefficient is significantly different than zero. All of the terms in our model have significant P-values. Lower 95% and Upper 95% columnsöa 95% confidence interval on the coefficient. If the confidence interval does not include zero, we will fail to reject the hypothesis that the coefficient is zero. None of the intervals in our example include zero. CORRELATION ANALYSIS As mentioned earlier, a correlation problem considers the joint variation of two variables, neither of which is restricted by the experimenter. Unlike regression analysis, which considers the effect of the independent variable(s) on a dependent variable, correlation analysis is concerned with the joint variation of one independent variable with another. In a correlation problem, the analyst has two measurements for each individual item in the sample. Unlike a regression study where the analyst controls the values of the x variables, correlation studies usually involve spontaneous variation in the variables being studied. Correlation methods for determining the strength of the linear relationship between two or more variables are among the most widely applied statistical

9 Regression and correlation analysis 513 techniques. More advanced methods exist for studying situations with more than two variables (e.g., canonical analysis, factor analysis, principal components analysis, etc.), however, with the exception of multiple regression, our discussion will focus on the linear association of two variables at a time. In most cases, the measure of correlation used by analysts is the statistic r, sometimes referred to as Pearson s product-moment correlation. Usually x and y are assumed to have a bivariate normal distribution. Under this assumption r is a sample statistic which estimates the population correlation parameter. One interpretation of r is based on the linear regression model described earlier, namely that r 2 is the proportion of the total variability in the y data which can be explained by the linear regression model. The equation for r is: r ¼ s xy s x s y ¼ n P xy P x P y pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½n P x 2 ð P xþ 2 Š½n P y 2 ð P yþ 2 Š ð14:7þ and, of course, r 2 is simply the square of r. r is bounded at 1 and +1. When the assumptions hold, the signi cance of r is tested by the regression ANOVA. Interpreting r can become quite tricky, so scatter plots should always be used (see above). When the relationship between x and y is non-linear, the explanatory power of r is difficult to interpret in precise terms and should be discussed with great care. While it is easy to see the value of very high correlations such as r ¼ 0:99, it is not so easy to draw conclusions from lower values of r, even when they are statistically significant (i.e., they are significantly different than 0.0). For example, r ¼ 0:5 does not mean the data show half as much clustering as a perfect straight-line fit. In fact, r ¼ 0doesnot mean that there is no relationship between the x and y data, as Figure shows. When r > 0, y tends to increase when x increases. When r < 0, y tends to decrease when x increases. Although r ¼ 0, the relationship between x and y is perfect, albeit non-linear. At the other extreme, r ¼ 1, a perfect correlation, does not mean that there is a cause and effect relationship between x and y. For example, both x and y might be determined by a third variable, z. In such situations, z is described as a lurking variable which hides in the background, unknown to the experimenter. Lurking variables are behind some of the infamous silly associations, such as the association between teacher s pay and liquor sales (the lurking variable is general prosperity).* *Itispossibletoevaluatetheassociationofxand y by removing the effect of the lurking variable. This can be done using regression analysis and computing partial correlation coefficients. This advanced procedure is described in most texts on regression analysis.

10 514 STATISTICAL ANALYSIS OF CAUSE AND EFFECT Figure Interpreting r ¼ 0 for curvilinear data. Establishing causation requires solid scientific understanding. Causation cannot be proven by statistics alone. Some statistical techniques, such as path analysis, can help determine if the correlations between a number of variables are consistent with causal assumptions. However, these methods are beyond the scope of this book. ANALYSIS OF CATEGORICAL DATA Chi-square, tables MAKING COMPARISONS USING CHI-SQUARE TESTS In Six Sigma, there are many instances when the analyst wants to compare the percentage of items distributed among several categories. The things might be operators, methods, materials, or any other grouping of interest. From each of the groups a sample is taken, evaluated, and placed into one of several categories (e.g., high quality, marginal quality, reject quality). The results can be presented as a table with m rows representing the groups of interest and k columns representing the categories. Such tables can be analyzed to answer the question Do the groups differ with regard to the proportion of items in the categories? The chi-square statistic can be used for this purpose.

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