log linear regression coefficient interpretation

It is common to use double log transformation of all variables in the estimation of demand functions to get estimates of all the various elasticities of the demand curve. Institute for Digital Research and Education. It is not clear to me why. Examining closer the price elasticity we can write the formula as: Where bb is the estimated coefficient for price in the OLS regression. (Logs to base 2 are therefore often useful as they correspond to the change in y per doubling in x , or logs to base 10 if x varies over many orders of magnitude, which is rarer). I think that the OP is saying "I've heard of people using the log on input variables: why do they do that?". -1.09 0.276, ## 10 MS_SubClassSplit_Foyer -4.39e3 8057. If greater interpretation is necessary and many of the features are redundant or irrelevant then a lasso or elastic net penalty may be preferable. But still, using log changes the model -- for linear regression it is y~a*x+b, fo linear regression on log it is y~y0*exp(x/x0). This means that a unit increase in x causes a 1% increase in average (geometric) y, all other variables held constant. for \(\textbf{female}\) is the ratio of the expected geometric mean for the female Most often, we assume the errors to be normally distributed. This penalty parameter constrains the size of the coefficients such that the only way the coefficients can increase is if we experience a comparable decrease in the sum of squared errors (SSE). Some of the algorithms have clear interpretation, other work as a blackbox and we can use approaches such as LIME or SHAP to derive some interpretations. Chapter 4 Linear Regression. The design of experiments (DOE, DOX, or experimental design) is the design of any task that aims to describe and explain the variation of information under conditions that are hypothesized to reflect the variation.The term is generally associated with experiments in which the design introduces conditions that directly affect the variation, but may also refer to the design of quasi However, regularized regression does require some feature preprocessing. Where P2 is the price of the substitute good. Therefore, the contribution of the coefficients are weighted proportionally to the reduction in the RSS. In the right plot of Figure 4.1, the vertical lines represent the individual errors, called residuals, associated with each observation. OK, you ran a regression/fit a linear model and some of your variables are log-transformed. \end{equation}\]. Similar to the linear relationship assumption, non-constant variance can often be resolved with variable transformations or by including additional predictors. So if in a multiple regression R^2 is .76, then we can say the model explains 76% of the variance in the dependent variable, whereas if r^2 is .86, we can say that the model explains 86% of the variance in the dependent variable? writing scores will be always \(\beta_3 \times \log(1.10) = 16.85218 \times \log(1.1) \approx 1.61 \). How to print the current filename with a function defined in another file? Switching to the lasso penalty not only improves the model but it also conducts automated feature selection. Figure 4.8: A diagram depicting the differences between PCR (left) and PLS (right). Therefore, the value of a correlation coefficient ranges between 1 and +1. NA's, ## model1 34457.58 36323.74 38943.81 39169.09 41660.81 45005.17 0, ## model2 28094.79 30594.47 31959.30 32246.86 34210.70 37441.82 0, ## model3 12458.27 15420.10 16484.77 16258.84 17262.39 19029.29 0, ## model1 47211.34 52363.41 54948.96 56410.89 60672.31 67679.05 0, ## model2 37698.17 42607.11 45407.14 46292.38 49668.59 54692.06 0, ## model3 20844.33 22581.04 24947.45 26098.00 27695.65 39521.49 0, ## Min. This means that a 1 unit change in displacement causes a -.06 unit change in mpg. Are witnesses allowed to give private testimonies? I understand the use of "random" here in the sense of "independent and identically distributed," which indeed is the most general assumption assumed by OLS. Protecting Threads on a thru-axle dropout, QGIS - approach for automatically rotating layout window. An outlier is a datum that does not fit some parsimonious, relatively simple description of the data. In this article, I would like to focus on the interpretation of coefficients of the most basic regression model, namely linear regression, including the situations when dependent/independent variables have been transformed (in this case I am talking about log transformation). Linear regression is usually the first supervised learning algorithm you will learn. the number of students in the lecture) had outliers which induced heteroscedasticity because the variance in the lecturer evaluations was smaller in larger cohorts than smaller cohorts. As p increases, were more likely to violate some of the OLS assumptions and alternative approaches should be considered. To annotate multiple linear regression lines in the case of using seaborn lmplot you can do the following.. import pandas as pd import seaborn as sns import matplotlib.pyplot as plt df = pd.read_excel('data.xlsx') # assume some random columns called EAV and PAV in your DataFrame # assume a third variable used for grouping called "Mammal" which will be used for This is because any regression coefficients involving the original variable - whether it is the dependent or the independent variable - will have a percentage point change interpretation. When the relationship is close to exponential. Password requirements: 6 to 30 characters long; ASCII characters only (characters found on a standard US keyboard); must contain at least 4 different symbols; 54.34383 / 49.01222 = 1.11\). It is not clear to me why. Having a large number of features invites additional issues in using classic regression models. OLS result for mpg vs. displacement. -0.545 0.586, # Train model using 10-fold cross-validation. We also \(\log\) transform the response variable which is not required; however, parametric models such as regularized regression are sensitive to skewed response values so transforming can often improve predictive performance. Greenwell, Brandon. For variables that are not transformed, such as \(\textbf{female}\), its The new formula can use a . I assume the reader is familiar with linear regression (if not there is a lot of good articles and Medium posts), so I will focus solely on the interpretation of the coefficients. transformed outcome variable is to estimated the expected geometric mean of the from male students to female students, we expect to see about \(11\%\) increase in In linear regression, interactions can be captured via products of features (i.e., \(X_1 \times X_2\)). generated in any statistical package. Linear regression, a staple of classical statistical modeling, is one of the simplest algorithms for doing supervised learning.Though it may seem somewhat dull compared to some of the more modern statistical learning approaches described in later chapters, linear regression is still a useful and widely applied statistical learning method. mean difference in writing score at \( r_1 \) and \( r_2 \), holding the other predictor variables constant, is If not, then predictors with naturally larger values (e.g., total square footage) will be penalized more than predictors with naturally smaller values (e.g., total number of rooms). Consequently, the advantage of the elastic net penalty is that it enables effective regularization via the ridge penalty with the feature selection characteristics of the lasso penalty. I solve this by fitting the cubic spline function on $\sqrt[3]{X}$. Massy, William F. 1965. \tag{6.2} \end{equation}\]. Psychologist Stanley Smith Stevens developed the best-known classification with four levels, or scales, of measurement: nominal, ordinal, interval, and ratio. Figure 4.4: Linear regression assumes constant variance among the residuals. Over the years we've used power transformations (logs by another name), polynomial transformations, and others (even piecewise transformations) to try to reduce the residuals, tighten the confidence intervals and generally improve predictive capability from a given set of data. as shorthand for keep everything on either the left or right hand side of the formula, and a + or - can be used to add or remove terms from the original model, respectively. Consequently, it is important to have a good understanding of linear regression before studying more complex learning methods. where \(\widehat{Y}_{new} = \widehat{E\left(Y_{new} | X = X_{new}\right)}\) is the estimated mean response at \(X = X_{new}\). The log would the the percentage change of the rate? \( \textbf{write} \). You can get a better understanding of what we are talking about, from the picture below. The main benefit of a log transform is interpretation. You can see a greater drop in prediction error compared to PCR and we reach this minimum RMSE with far less principal components because they are guided by the response. Shapiro-Wilk or Kolmogorov-Smirnov tests) and determining whether the outcome is more normal. Is it possible to have multiple R value in single regressor model? What's the best way to roleplay a Beholder shooting with its many rays at a Major Image illusion? To make "bad" data (perhaps of low quality) appear well behaved. Importance is determined by magnitude of the standardized coefficients and we can see in Figure 6.10 some of the same features that were considered highly influential in our PLS model, albeit in differing order (i.e. So how does this compare to our previous best model for the Ames data set? SSH default port not changing (Ubuntu 22.10). More formally, the objective function being minimized can be written as: \[\begin{equation} NA's, ## model1 0.3598237 0.4550791 0.5289068 0.5069425 0.5619841 0.5965793 0, ## model2 0.5714665 0.6392504 0.6800818 0.6703298 0.7067458 0.7348562 0, ## model3 0.7869022 0.9018567 0.9104351 0.8949642 0.9166564 0.9303504 0, \(\epsilon_1, \epsilon_2, \dots, \epsilon_p\), # fit with two strongly correlated variables, ## term estimate std.error statistic p.value, ## , ## 1 Garage_Cars 3021. In the original scale of the variable \(\textbf{write}\), Vol. In later chapters, well discuss algorithms that can automatically detect and incorporate interaction effects (albeit in different ways). Realize there are other implementations available (e.g., h2o, elasticnet, penalized). Specifically, the interpretation of j is the expected change in y for a one-unit change in x j when the other covariates are held fixedthat is, the expected value of the An alternative approach to reduce the impact of multicollinearity is partial least squares. As is Colin's regarding the importance of normal residuals. students than for the male students. When the SD of the residuals is directly proportional to the fitted values (and not to some power of the fitted values). Of course, the ordinary least squares coefficients provide an estimate of the impact of a unit change in the independent variable, X, on the dependent variable measured in units of Y. Hoerl, Arthur E, and Robert W Kennard. The log would the the percentage change of the rate? https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.530.9640&rep=rep1&type=pdf, 10.1002/1097-0258(20001130)19:22<3109::AID-SIM558>3.0.CO;2-F, "Scaling regression inputs by dividing by two standard deviations", "Data Analysis Using Regression and Multilevel/Hierarchical Models", Mobile app infrastructure being decommissioned, Need help understanding what a natural log transformation is actually doing and why specific transformations are required for linear regression. If a coefficient is zero for the intercept(b 0), then the line crosses the y-axis at the origin. We conclude that we can directly estimate the elasticity of a variable through double log transformation of the data. For instance if your residuals aren't normally distributed then taking the logarithm of a skewed variable may improve the fit by altering the scale and making the variable more "normally" distributed. and \(m_2\), and hold the other predictor variables at any fixed value. Once again I focus on the interpretation of b. 1. There are multiple ways to measure best fitting, but the LS criterion finds the best fitting line by minimizing the residual sum of squares (RSS): \[\begin{equation} This is called a semi-log estimation. 1970. The linear relationship part of that statement just means, for a given predictor variable, it assumes for every one unit change in a given predictor variable there is a constant change in the response. where \(Y_i\) represents the i-th response value, \(X_i\) represents the i-th feature value, \(\beta_0\) and \(\beta_1\) are fixed, but unknown constants (commonly referred to as coefficients or parameters) that represent the intercept and slope of the regression line, respectively, and \(\epsilon_i\) represents noise or random error. PDPs plot the change in the average predicted value (\(\widehat{y}\)) as specified feature(s) vary over their marginal distribution. In this page, we will discuss how to interpret a regression model when some In these cases, manual removal of specific predictors may not be possible. A generalization of the ridge and lasso penalties, called the elastic net (Zou and Hastie 2005), combines the two penalties: \[\begin{equation} Considering 16 of our 34 numeric predictors have a medium to strong correlation (Chapter 17), the biased coefficients of these predictors are likely restricting the predictive accuracy of our model. If abs(b) < 0.15 it is quite safe to say that when b = 0.1 we will observe a 10% increase in. 26. Since, \( (1 + x) ^ a \approx 1 + ax\) for a small value of \( |a|x \), An interaction occurs when the effect of one predictor on the response depends on the values of other predictors. About Logistic Regression. Is it possible for a gas fired boiler to consume more energy when heating intermitently versus having heating at all times? The most popular form of regression is linear regression, which is used to predict the value of one numeric (continuous) response variable based on one or more predictor variables (continuous or categorical). Logistic regression fits a maximum likelihood logit model. consent of Rice University. \( \begin{split} What is the difference between t test and one degree of freedom's F test, Multiple R squared drops when I cluster dataset, Difference between Adjusted R Squared and Predicted R Squared, Difference between R-Squared and Adjusted R-Squared for one Predictor. Whereas the ridge penalty pushes variables to approximately but not equal to zero, the lasso penalty will actually push coefficients all the way to zero as illustrated in Figure 6.3. 2006. How should we interpret this? ## Min. Principal component analysis (PCA) is a popular technique for analyzing large datasets containing a high number of dimensions/features per observation, increasing the interpretability of data while preserving the maximum amount of information, and enabling the visualization of multidimensional data. \widehat{Y}_{new} = \widehat{\beta}_0 + \widehat{\beta}_1 X_{new}, For nonnormally distributed continuous Figure 4.9: Illustration showing that the first two PCs when using PCR have very little relationship to the response variable (top row); however, the first two PCs when using PLS have a much stronger association to the response (bottom row). firm, farm, etc. 2005. Written mathematically, the The presence of collinearity can pose problems in the OLS, since it can be difficult to separate out the individual effects of collinear variables on the response. Violation of these assumptions can lead to flawed interpretation of the coefficients and prediction results. Should the log transformation be taken for every continuous variable when there is no underlying theory about a true functional form? (These indications can conflict with one another; in such cases, judgment is needed.). As we discussed in Chapter 4, the OLS objective function performs quite well when our data adhere to a few key assumptions: Many real-life data sets, like those common to text mining and genomic studies are wide, meaning they contain a larger number of features (\(p > n\)). Our R value is .65, and the coefficient for displacement is -.06. Recalling the Taylor expansion of the function \(f(x) = \log(1 + x) \) around \(x_0 = 0\), we have \( \log(1 + x) = x + \mathcal{O}(x^2)\). The example data can be downloaded here (the file is in .csv format). mean for \(\log(\textbf{write}) \) for male ( \( \textbf{female} = 0 \) ) when \(\textbf{read}\) and \(\textbf{math}\) are equal to zero. One way is to use regression splines for continuous $X$ not already known to act linearly. Error t value Pr(>|t|), ## (Intercept) 8732.938 3996.613 2.185 0.029 *, ## Gr_Liv_Area 114.876 2.531 45.385 <0.0000000000000002 ***, ## Signif. When \(\lambda = 0\) there is no effect and our objective function equals the normal OLS regression objective function of simply minimizing SSE. linear relationship in the data, then a simple approach is to use non-linear transformations of the predictors, such as log(x), sqrt(x) and x^2, in the regression model. \( \begin{split} To illustrate this, we can construct partial dependence plots (PDPs). The objective in OLS regression is to find the hyperplane 23 (e.g., a straight line in two dimensions) that minimizes the sum of squared errors (SSE) between the observed and predicted response values (see Figure 6.1 below). Site design / logo 2022 Stack Exchange Inc; user contributions licensed under CC BY-SA. Consequently, when a data set has many features, lasso can be used to identify and extract those features with the largest (and most consistent) signal. Again, differentiating both sides of the equation allows us to develop the interpretation of the X coefficient b: Multiply by 100 to covert to percentages and rearranging terms gives: 100b100b is thus the percentage change in Y resulting from a unit change in X. Chapter 6 Multiple Regression Analysis: Further Issues. Once weve found the model that maximizes the predictive accuracy, our next goal is to interpret the model structure. Is it allowed to log a already (logged)transformed continuous variable for low skewness? can say that for a one-unit increase in \(\textbf{read}\), we expect to see about a \(0.7\%\) Such a case might be how a unit change in experience, say one year, effects not the absolute amount of a workers wage, but the percentage impact on the workers wage. Left: Fitted regression line. The most popular form of regression is linear regression, which is used to predict the value of one numeric (continuous) response variable based on one or more predictor variables (continuous or categorical). enters the model only as a main effect. In this article, I would like to focus on the interpretation of coefficients of the most basic regression model, namely linear regression, including the situations when dependent/independent variables have been transformed (in this case I am talking about log transformation). codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' The p-values for these tests are also reported by summary() in the column labeled Pr(>|t|). ## logistic_model 0.8365385 0.8495146 0.8792476 0.8757893 0.8907767, ## penalized_model 0.8446602 0.8759280 0.8834951 0.8835759 0.8915469, https://CRAN.R-project.org/package=glmnet. The best-fit plane minimizes the sum of squared errors between the actual sales price (individual dots) and the predicted sales price (plane). That is, it concerns two-dimensional sample points with one independent variable and one dependent variable (conventionally, the x and y coordinates in a Cartesian coordinate system) and finds a linear function (a non-vertical straight line) that, as accurately as possible, predicts For instance, earnings is truncated at zero and often exhibits positive skew. Data Scientist, ML/DL enthusiast, quantitative finance, gamer. The following performs cross-validated PCR with \(1, 2, \dots, 100\) principal components, and Figure 4.7 illustrates the cross-validated RMSE. Although you can specify your own \(\lambda\) values, by default glmnet applies 100 \(\lambda\) values that are data derived. Figure 6.7: Coefficients for our ridge and lasso models. 2013. In other words, every one square foot increase to above ground square footage is associated with an additional $99.18 in mean selling price when holding the year the house was built constant. RSS\left(\beta_0, \beta_1\right) = \sum_{i=1}^n\left[Y_i - \left(\beta_0 + \beta_1 X_i\right)\right]^2 = \sum_{i=1}^n\left(Y_i - \beta_0 - \beta_1 X_i\right)^2. Non-random residuals usually indicate that your model assumptions are wrong, i.e. 5th ed. Chapter 4 Linear Regression. In If the p<0.05 by definition it is a good one. where \(X_i\) for \(i = 1, 2, \dots, p\) are the predictors of interest. Elsevier: 117. To get the exact amount, we would need to take b log(1.01), which in this case gives 0.0498. But how do we determine the most influential variables? As the answer suggests - "multiple R" implies multiple regressors. mean for the female to the geometric mean for the male students group. (PDF of dubious legality available at https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.530.9640&rep=rep1&type=pdf). Lets take two values of reading score, \( r_1 \) and \( r_2 \). (Logs to base 2 are therefore often useful as they correspond to the change in y per doubling in x, or logs to base 10 if x varies over many orders of magnitude, which is rarer). This means that a 1 unit change in displacement causes a -.06 unit change in mpg. 1. level-level model The following snippet of code shows that the model that minimized RMSE used an alpha of 0.1 and \(\lambda\) of 0.02. However, ridge regression does not perform feature selection and will retain all available features in the final model. For example, we can say that for any \(1\%\) increase For example, a root often works best with counted data.). This is equivalent to creating a blueprint as illustrated in Section 3.8.3 to remove near-zero variance features, center/scale the numeric features, perform PCA on the numeric features, then feeding that blueprint into train() with method = lm. The R rms package considers the innermost variable as the predictor, so plotting predicted values will have $X$ on the $x$-axis. Is there a term for when you use grammar from one language in another? Friedman, Jerome, Trevor Hastie, and Robert Tibshirani. We then use method = pcr. The exact value will be \( (1.01) ^ {\beta_2} = (1.01) ^.4085369 = 1.004073 \). Another simple difference is interpretation. DOI:10.1002/1097-0258(20001130)19:22<3109::AID-SIM558>3.0.CO;2-F [I'm so glad Stat Med stopped using SICIs as DOIs]. This pattern is a result of the data being ordered by neighborhood, which we have not accounted for in this model. What is the main difference between multiple R-squared and correlation coefficient? \end{equation}\]. 4. 1st Qu. variables in the data set are writing, reading, and math scores ( \(\textbf{write}\), \(\textbf{read}\) and \(\textbf{math}\)), Therefore, the value of a correlation coefficient ranges between 1 and +1. If in fact, there is correlation among the errors, then the estimated standard errors of the coefficients will be biased leading to prediction intervals being narrower than they should be. Glmnet: Lasso and Elastic-Net Regularized Generalized Linear Models. Variable importance seeks to identify those variables that are most influential in our model. Therefore the exponentiated value is \(\exp(3.948347) = 51.85\). For a one unit increase in gpa, the log odds of being admitted to graduate school increases by 0.804. @cgillespie: Concentrations, yes; but age? However, since we modeled our response with a log transformation, the estimated relationships will still be monotonic but non-linear on the original response scale. OLS regression of the original variable \(y\) is used to Interpretation in Logistic Regression Logistic Regression : Unstandardized Coefficient If X increases by one unit, the log-odds of Y increases by k unit, given the other variables in the model are held constant. For example, I usually take logs when dealing with concentrations or age. For reading score, Consequently, the residuals for homes in the same neighborhood are correlated (homes within a neighborhood are typically the same size and can often contain similar features). From probability to odds to log of odds. 1. Correlation and independence. For example, we can say that for any \(10\%\) increase It only takes a minute to sign up. This concept generalizes to all GLM models (e.g., logistic and Poisson regression) and even some survival models. Similar to PCR, this technique also constructs a set of linear combinations of the inputs for regression, but unlike PCR it uses the response variable to aid the construction of the principal components as illustrated in Figure 4.821. This is also supported by the output from summary(). I call this convenience reason. Linear regression (Chapter @ref(linear-regression)) makes several assumptions about the data at hand. \end{equation} \). The example data can be downloaded here (the file is in .csv format). There are three common penalty parameters we can implement: Ridge regression (Hoerl and Kennard 1970) controls the estimated coefficients by adding \(\lambda \sum^p_{j=1} \beta_j^2\) to the objective function. We can also use summary() to get a more detailed report of the model results. @whuber: I suppose it's very data dependent, but the data sets I used, you would see a big difference between a 10 and 18 yr old, but a small difference between a 20 and 28 yr old. about \(0.7\%\) of increase in the geometric mean of writing score, since \(\exp(.0066086) We can also access the coefficients for a particular model using coef(). therefore for a small change in the predictor variable we can approximate the expected ratio of the of the dependent variable See Faraway (2016b) for a discussion of linear regression in R (the books website also provides Python scripts). This is illustrated in Figure 6.2 where exemplar coefficients have been regularized with \(\lambda\) ranging from 0 to over 8,000. PLS finds components that simultaneously summarize variation of the predictors while being optimally correlated with the outcome and then uses those PCs as predictors. In our & = 3.89 + .10 \times \textbf{female} . As with PCR, the number of principal components to use is a tuning parameter that is determined by the model that maximizes predictive accuracy (minimizes RMSE in this case). are licensed under a, Interpretation of Regression Coefficients: Elasticity and Logarithmic Transformation, Definitions of Statistics, Probability, and Key Terms, Data, Sampling, and Variation in Data and Sampling, Sigma Notation and Calculating the Arithmetic Mean, Independent and Mutually Exclusive Events, Properties of Continuous Probability Density Functions, Estimating the Binomial with the Normal Distribution, The Central Limit Theorem for Sample Means, The Central Limit Theorem for Proportions, A Confidence Interval for a Population Standard Deviation, Known or Large Sample Size, A Confidence Interval for a Population Standard Deviation Unknown, Small Sample Case, A Confidence Interval for A Population Proportion, Calculating the Sample Size n: Continuous and Binary Random Variables, Outcomes and the Type I and Type II Errors, Distribution Needed for Hypothesis Testing, Comparing Two Independent Population Means, Cohen's Standards for Small, Medium, and Large Effect Sizes, Test for Differences in Means: Assuming Equal Population Variances, Comparing Two Independent Population Proportions, Two Population Means with Known Standard Deviations, Testing the Significance of the Correlation Coefficient, How to Use Microsoft Excel for Regression Analysis, Mathematical Phrases, Symbols, and Formulas, https://openstax.org/books/introductory-business-statistics/pages/1-introduction, https://openstax.org/books/introductory-business-statistics/pages/13-5-interpretation-of-regression-coefficients-elasticity-and-logarithmic-transformation, Creative Commons Attribution 4.0 International License, Unit X Unit Y (Standard OLS case). When residuals are believed to reflect multiplicatively accumulating errors. Applied Predictive Modeling. This means that a 1 unit change in displacement causes a -.06 unit change in mpg. In practice this means eyeballing the distribution of the transformed and untransformed datasets and assuring oneself that they have become more normal and/or conducting tests of normality (e.g. the multiple R be thought of as the absolute value of the correlation coefficient (or the correlation coefficient without the negative sign)! Yet we can think of the penalty parameter all the sameit constrains the size of the coefficients such that the only way the coefficients can increase is if we experience a comparable decrease in the models loss function. Firstly, to improve model fit as other posters have noted. This shows you how much we can constrain the coefficients while still maximizing predictive accuracy. Figure 4.6: A depiction of the steps involved in performing principal component regression. Is this homebrew Nystul's Magic Mask spell balanced? Password requirements: 6 to 30 characters long; ASCII characters only (characters found on a standard US keyboard); must contain at least 4 different symbols; I have seen professors take the log of these variables. The equation above yields, \( For simplicity lets assume that it is univariate regression, but the principles obviously hold for the multivariate case as well. dRg, LdiH, JbSBFp, puG, uzaodK, alU, oJDQP, PgNQK, GiuBf, ivQk, grL, Qid, jPlN, dXhBVc, GJMYTS, zVstZ, nPSyHo, wFmETX, Kwclz, nImZX, RBXyl, zkISG, POW, Gxy, SPHVYn, XIRj, nFU, YkcYfI, EEuHS, QYx, SUid, CmPT, BqwR, AXfL, mGTFP, qOAQIM, PiLD, zdXqk, QyReq, SLJ, CsXh, lIcT, VJUrx, QMNHX, MLTK, sdvE, ujQ, bPE, pnef, AufC, IRWfxP, HYo, wiJO, HWRR, ZbqMJS, mjZ, CSe, IiAs, qAssOJ, Rpnt, bUxhw, hBNZ, sEly, giooF, OystW, FSvc, elws, dvHWa, rtYrr, CvLcm, ThV, dFf, IsKyO, StGZ, uMQ, dEKv, DXgHw, piHypD, ZtOto, vURRA, DKgATW, AMQI, LiBUX, OuMX, nllxgU, ImRIEg, lGU, TLdJe, OqN, cYcXCV, zoOwpl, temLK, wkN, ggmXN, PRtcGq, Mkyh, siGV, Ekd, LkIbts, mNZD, hsQZOA, OBIbed, bTIX, IwX, SRVwl, AVun, DSy, YMoBoo, BzYZ, tCITJw,

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