# how to calculate b1 and b2 in multiple regression

Simply stated, when comparing two models used to predict the same response variable, we generally prefer the model with the higher value of adjusted $$R^2$$ see Lesson 10 for more details. 24. On this occasion, Kanda Data will write a tutorial on manually calculating the coefficients bo, b1, b2, and the coefficient of determination (R Squared) in multiple linear regression. .ai-viewport-0 { display: none !important;} color: #747474; .main-navigation ul li ul li:hover > a, You also have the option to opt-out of these cookies. Skill Development From the above given formula of the multi linear line, we need to calculate b0, b1 and b2 . (function(){var o='script',s=top.document,a=s.createElement(o),m=s.getElementsByTagName(o),d=new Date(),timestamp=""+d.getDate()+d.getMonth()+d.getHours();a.async=1;a.src='https://cdn4-hbs.affinitymatrix.com/hvrcnf/wallstreetmojo.com/'+ timestamp + '/index?t='+timestamp;m.parentNode.insertBefore(a,m)})(); Adjusted $$R^2=1-\left(\frac{n-1}{n-p}\right)(1-R^2)$$, and, while it has no practical interpretation, is useful for such model building purposes. I have read the econometrics book by Koutsoyiannis (1977). .cat-links a, The calculations of b0, b1, and b2 that I have calculated can be seen in the image below: Furthermore, the results of calculations using the formula obtained the following values: To crosscheck the calculations, I have done an analysis using SPSS with the estimated coefficients as follows: Well, thats the tutorial and discussion this time I convey to you. For further procedure and calculation, refer to the: Analysis ToolPak in Excel article. . For the audio-visual version, you can visit the KANDA DATA youtube channel. .woocommerce a.button.alt, background-color: #cd853f; #bbpress-forums .bbp-topics a:hover { Multiple Linear Regression Calculator Multiple regression formulas analyze the relationship between dependent and multiple independent variables. } Learning Objectives Contd 6. In the formula, n = sample size, p = number of parameters in the model (including the intercept) and SSE = sum of squared errors. In this particular example, we will see which variable is the dependent variable and which variable is the independent variable. Answer (1 of 4): I am not sure what type of answer you want: it is possible to answer your question with a bunch of equations, but if you are looking for insight, that may not be helpful. You can now share content with a Team. Suppose we have the following dataset with one response variabley and two predictor variables X1 and X2: Use the following steps to fit a multiple linear regression model to this dataset. Use the following steps to fit a multiple linear regression model to this dataset. Sports Direct Discount Card, +91 932 002 0036 How to calculate b0 (intercept) and b1, b2. background-color: #dc6543; Y=b0+b1*x1+b2*x2 where: b1=Age coefficient b2=Experience coefficient #use the same b1 formula(given above) to calculate the coefficients of Age and Experience Multiple regression analysis is a statistical technique that analyzes the relationship between two or more variables and uses the information to estimate the value of the dependent variables. Analytics Vidhya is a community of Analytics and Data Science professionals. ::selection { .screen-reader-text:active, To find b2, use the formula I have written in the previous paragraph. This simple multiple linear regression calculator uses the least squares method to find the line of best fit for data comprising two independent X values and one dependent Y value, allowing you to estimate the value of a dependent variable (Y) from two given independent (or explanatory) variables (X 1 and X 2). } This model generalizes the simple linear regression in two ways. .main-navigation ul li.current-menu-item a, .woocommerce .woocommerce-message:before { document.getElementById( "ak_js_1" ).setAttribute( "value", ( new Date() ).getTime() ); Copyright 2023 . are known (they can be calculated from the sample data values). number of bedrooms in this case] constant. Correlations among the predictors can change the slope values dramatically from what they would be in separate simple regressions. The estimate of 1 is obtained by removing the effects of x2 from the other variables and then regressing the residuals of y against the residuals of x1. For instance, suppose that we have three x-variables in the model. The data that researchers have collected can be seen in the table below: Following what I have written in the previous paragraph, to avoid errors in calculating manually, I am here using Excel. color: #cd853f; border: 2px solid #CD853F ; In the simple linear regression case y = 0 + 1x, you can derive the least square estimator 1 = ( xi x) ( yi y) ( xi x)2 such that you don't have to know 0 to estimate 1. Central Building, Marine Lines, .ld_newsletter_640368d8e55e4.ld-sf input{font-family:avenirblook!important;font-weight:400!important;font-style:normal!important;font-size:18px;}.ld_newsletter_640368d8e55e4.ld-sf .ld_sf_submit{font-family:avenirblook!important;font-weight:400!important;font-style:normal!important;font-size:18px;}.ld_newsletter_640368d8e55e4.ld-sf button.ld_sf_submit{background:rgb(247, 150, 34);color:rgb(26, 52, 96);} */ .light-color:hover, border-color: #dc6543; Hopefully, it will provide a deeper understanding for you. Error rate This is small negligible value also known as epsilon value. .top-header .widget_contact ul li a:hover, In this case, the data used is quarterly time series data from product sales, advertising costs, and marketing staff. window['GoogleAnalyticsObject'] = 'ga'; Any feedback is most welcome. Now lets move on to consider a regression with more than one predictor. } Odit molestiae mollitia margin-top: 30px; b 0 and b 1 are called point estimators of 0 and 1 respectively. Our Methodology display: block !important; .entry-title a:active, For the above data, If X = 3, then we predict Y = 0.9690 If X = 3, then we predict Y =3.7553 If X =0.5, then we predict Y =1.7868 2 If we took the averages of estimates from many samples, these averages would approach the true Here we need to be careful about the units of x1. Relative change shows the change of a value of an indicator in the first period and in percentage terms, i.e. .tag-links, B 1 = b 1 = [ (x. i. After we have compiled the specifications for the multiple linear regression model and know the calculation 888+ PhD Experts 9.3/10 Quality score Suppose you have predictor variables X1, X2, and X3 and. It may well turn out that we would do better to omit either $$x_1$$ or $$x_2$$ from the model, but not both. For this calculation, we will not consider the error rate. Introduction to Statistics is our premier online video course that teaches you all of the topics covered in introductory statistics. Save my name, email, and website in this browser for the next time I comment. Calculation of Multiple Regression with Three Independent Variables Using a Programable Pocket Calculator By: : Paul D. Evenson Assoc. background-color: #CD853F ; } ul.default-wp-page li a { color: white; b0 is constant. B0 b1 b2 calculator - The easy-to-use simple linear regression calculator gives you step-by-step solutions to the estimated regression equation, coefficient of. .widget ul li a } @media screen and (max-width:600px) { B0 is the intercept, the predicted value of y when the x is 0. The additional columns are adjusted to the components of the calculation formulas b0, b1, and b2. Linear regression calculator Exercises for Calculating b0, b1, and b2. { .site-info .social-links a{ Hakuna Matata Animals, Edit Report an issue 30 seconds. 1 pt. background-color: #cd853f; It can be manually enabled from the addins section of the files tab by clickingon manage addins, andthen checkinganalysis toolpak.read more article. Edit Report an issue 30 seconds. position: absolute; Learn more about us. Let us try and understand the concept of multiple regression analysis with the help of an example. { padding-bottom: 0px; Each $$\beta$$ parameter represents the change in the mean response, E(, For example, $$\beta_1$$ represents the estimated change in the mean response, E(, The intercept term, $$\beta_0$$, represents the estimated mean response, E(, Other residual analyses can be done exactly as we did in simple regression. .woocommerce #respond input#submit, This category only includes cookies that ensures basic functionalities and security features of the website. Regression Calculations yi = b1 xi,1 + b2 xi,2 + b3 xi,3 + ui The q.c.e. } .go-to-top a:hover margin-left: auto; background: #cd853f; x1,x2,,xn). a, .main-navigation ul li.current-menu-item ul li a:hover, You can use this formula: Y = b0 + b1X1 + b1 + b2X2 + . Contact To carry out the test, statistical software will report p-values for all coefficients in the model. (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start': As you can see to calculate b0, we need to first calculate b1 and b2. Here, what are these coefficient, and how to choose coefficient values? [wbcr_snippet id="84501"] Regression Parameters. where a, the intercept, = (Y - b (X)) / N. with multiple regression, the formula is: Y=a + b1X1 + b2X2 + b3X3, etc. Multiple regression formulas analyze the relationship between dependent and multiple independent variables. margin-top: 0px; Then test the null of = 0 against the alternative of . Save my name, email, and website in this browser for the next time I comment. } Mob:+33 699 61 48 64. .slider-buttons a { .woocommerce input.button.alt, .ld_button_640368d8ef2ef.btn-icon-solid .btn-icon{background:rgb(247, 150, 34);}.ld_button_640368d8ef2ef.btn-icon-circle.btn-icon-ripple .btn-icon:before{border-color:rgb(247, 150, 34);}.ld_button_640368d8ef2ef{background-color:rgb(247, 150, 34);border-color:rgb(247, 150, 34);color:rgb(26, 52, 96);}.ld_button_640368d8ef2ef .btn-gradient-border defs stop:first-child{stop-color:rgb(247, 150, 34);}.ld_button_640368d8ef2ef .btn-gradient-border defs stop:last-child{stop-color:rgb(247, 150, 34);} } Let us try and understand the concept of multiple regression analysis with the help of another example. '&l='+l:'';j.async=true;j.src= footer a:hover { @media screen and (max-width:600px) { Arcu felis bibendum ut tristique et egestas quis: $$\begin{equation} y_{i}=\beta_{0}+\beta_{1}x_{i,1}+\beta_{2}x_{i,2}+\ldots+\beta_{p-1}x_{i,p-1}+\epsilon_{i}. The formula will consider the weights assigned to each category. .slider-buttons a { Necessary cookies are absolutely essential for the website to function properly. margin-bottom: 0; color: #fff; To make it easier to practice counting, I will give an example of the data I have input in excel with n totaling 15, as can be seen in the table below: To facilitate calculations and avoid errors in calculating, I use excel. Then test the null of = 0 against the alternative of < 0. info@degain.in The multiple independent variables are chosen, which can help predict the dependent variable to predict the dependent variable. Support Service } The regression analysis helps in the process of validating whether the predictor variables are good enough to help in predicting the dependent variable. It can be manually enabled from the addins section of the files tab by clickingon manage addins, andthen checkinganalysis toolpak. Next, based on the formula presented in the previous paragraph, we need to create additional columns in excel. color: #747474; multiple regression up in this way, b0 will represent the mean of group 1, b1 will represent the mean of group 2 - mean of group 1, and b2 will represent the mean of group 3 - mean of group 1. } .main-navigation ul li.current-menu-item ul li a:hover, } background-color: #747474 !important; Sign up to get the latest news .ai-viewport-3 { display: none !important;} background-color: #747474; { Sports Direct Discount Card, basic equation in matrix form is: y = Xb + e where y (dependent variable) is (nx1) or ( . Despite its popularity, interpretation of the regression coefficients of any but the simplest models is sometimes, well.difficult. .ai-viewport-1 { display: inherit !important;} \end{equation*}$$. /*! Bottom line on this is we can estimate beta weights using a correlation matrix. Regression Calculations yi = b1 xi,1 + b2 xi,2 + b3 xi,3 + ui The q.c.e. This page shows how to calculate the regression line for our example using the least amount of calculation. For how to manually calculate the estimated coefficients in simple linear regression, you can read my previous article entitled: Calculate Coefficients bo, b1, and R Squared Manually in Simple Linear Regression. In detail, it can be seen as follows: Based on what has been calculated in the previous paragraphs, we have manually calculated the coefficients of bo, b1 and the coefficient of determination (R squared) using Excel. ul li a:hover, For example, the equation Y represents the formula is equal to a plus bX1 plus cX2 plus dX3 plus E where Y is the dependent variable, and X1, X2, and X3 are independent variables. } In the formula. {color: #CD853F;} A boy is using art supplies. INTERCEPT (A1:A6,B1:B6) yields the OLS intercept estimate of 0.8. In matrix terms, the formula that calculates the vector of coefficients in multiple regression is: b = (X'X)-1 X'y In our example, it is = -6.867 + 3.148x 1 - 1.656x 2. Multiple Regression: Two Independent Variables Case Exercises for Calculating b0, b1, and b2. color: #CD853F ; +91 932 002 0036, Temp Staffing Company X Y i = nb 0 + b 1 X X i X X iY i = b 0 X X i+ b 1 X X2 2.This is a system of two equations and two unknowns. How do you interpret b1 in multiple linear regression. color: #CD853F ; input#submit { So when you call regression, call it as regression("b1", x, y) or regression("b0", x, y).. border: 1px solid #cd853f; .ai-viewport-2 { display: inherit !important;} How to calculate multiple linear regression. The higher R Squared indicates that the independent variables variance can explain the variance of the dependent variable well. } .main-navigation ul li:hover a, .widget_contact ul li a:hover, color: #cd853f; .el-pack .sow-headline { Completing these calculations requires an understanding of how to calculate using a mathematical equation formula. Before we find b1 and b2, we will compute the values for the following for both x1 and x2 so that we can compute b1 and b2 followed by b0: Here i stands for the value of x say variable 1 or variable 2 and N is the number of records which is 10 in this case. Next, you calculate according to the Excel tables formula. Multiple-choice . 'event': 'templateFormSubmission' This is a generalised regression function that fits a linear model of an outcome to one or more predictor variables. .btn-default:hover { .btn-default:hover, These cookies will be stored in your browser only with your consent. background: #cd853f; plays 130 questions New! Mumbai 400 002. Central Building, Marine Lines, .ai-viewport-1 { display: none !important;} The coefficients b1 and b2 are the unknowns, the values for cov(y1,x1), cov(x1,x2), etc. Linear regression is one of the most popular statistical techniques. b0 and b1 don't exist when you call the function, so you can't pass them in as arguments---you can pass them in as strings, which is what switch expects. .search-submit Ok, this is the article I can write for you. Please note: The categorical value should be converted to ordinal scale or nominal assigning weights to each group of the category. The regression formulaRegression FormulaThe regression formula is used to evaluate the relationship between the dependent and independent variables and to determine how the change in the independent variable affects the dependent variable. It is part 1 of 3 part. .sow-carousel-title a.sow-carousel-next { .main-navigation ul li.current-menu-item ul li a:hover { window['ga'] = window['ga'] || function() { } From the above given formula of the multi linear line, we need to calculate b0, b1 and b2 . /* Therefore, because the calculation is conducted manually, the accuracy in calculating is still prioritized. border: 1px solid #cd853f; Xi2 = independent variable (Weight in Kg) B0 = y-intercept at time zero. This article has been a guide to the Multiple Regression Formula. Loan Participation Accounting, Follow us Based on the formula I wrote in the previous paragraph, finding the Intercept Estimation Coefficient (b0) can be seen as follows: R Squared in multiple linear regression shows the goodness of fit of a model. } 2. }. border: 1px solid #cd853f; border-top: 2px solid #CD853F ; It is "r = n (xy) x y / [n* (x2 (x)2)] * [n* (y2 (y)2)]", where r is the Correlation coefficient, n is the number in the given dataset, x is the first variable in the context and y is the second variable. h4 { .tag-links, color: #dc6543; as well as regression coefficient value (Rsquare)? color: #dc6543; Note that the hypothesized value is usually just 0, so this portion of the formula is often omitted. and the intercept (b0) can be calculated as. Refer to the figure below. { For our example above, the t-statistic is: $$\begin{equation*} t^{*}=\dfrac{b_{1}-0}{\textrm{se}(b_{1})}=\dfrac{b_{1}}{\textrm{se}(b_{1})}. Y = a + b X +read more for the above example will be. sinners in the hands of an angry god hyperbole how to calculate b1 and b2 in multiple regression. 2 from the regression model and the Total mean square is the sample variance of the response ( sY 2 2 is a good estimate if all the regression coefficients are 0). .main-navigation ul li.current-menu-ancestor a, Out of these cookies, the cookies that are categorized as necessary are stored on your browser as they are essential for the working of basic functionalities of the website. } This tutorial explains how to perform multiple linear regression by hand. 71. } .cat-links, Two-Variable Regression. .entry-meta .entry-format:before, .main-navigation ul li ul li a:hover, Save my name, email, and website in this browser for the next time I comment. It is possible to estimate just one coefficient in a multiple regression without estimating the others. We wish to estimate the regression line y = b1 + b2*x Do this by Tools / Data Analysis / Regression. b0 = -6.867. Suppose we have the following dataset with one response variable, The estimated linear regression equation is: =b, Here is how to interpret this estimated linear regression equation: = -6.867 + 3.148x, An Introduction to Multivariate Adaptive Regression Splines. 'https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f); Correlation is a statistical measure between two variables that is defined as a change in one variable corresponding to a change in the other. It is mandatory to procure user consent prior to running these cookies on your website. .ai-viewport-1 { display: none !important;} Next, please copy and paste the formula until you get the results as shown in the image below: To find b1, use the formula I have written in the previous paragraph. Then we would say that when square feet goes up by 1, then predicted rent goes up by 2.5. But first, we need to calculate the difference between the actual data and the average value. Sign up to get the latest news (function(w){"use strict";if(!w.loadCSS){w.loadCSS=function(){}} For this example, Adjusted R-squared = 1 - 0.65^2/ 1.034 = 0.59. Relative change is calculated by subtracting the value of the indicator in the first period from the value of the indicator in the second period which is then divided by the value of the indicator in the first period and the result is taken out in percentage terms. The formula used to calculate b0, b1 and b2 based on the book Koutsoyiannis (1977) can be seen as follows: Calculating the values of b0, b1 and b2 cannot be conducted simultaneously. Step 1: Calculate X12, X22, X1y, X2y and X1X2. Degain manages and delivers comprehensive On-site Service Solutions that proactively preserve the value of each property, process, and products. .dpsp-share-text { In the equation, y is the single dependent variable value of which depends on more than one independent variable (i.e. While running this analysis, the main purpose of the researcher is to find out the relationship between the dependent and independent variables. How do you calculate b1 in regression? background-color: #CD853F ; By closing this banner, scrolling this page, clicking a link or continuing to browse otherwise, you agree to our Privacy Policy, You can see how this popup was set up in our step-by-step guide: https://wppopupmaker.com/guides/auto-opening-announcement-popups/. This would be interpretation of b1 in this case. In many applications, there is more than one factor that inuences the response. R Squared formula depicts the possibility of an event's occurrence within an expected outcome. border: 1px solid #cd853f; Step 2: Calculate Regression Sums. The dependent variable in this regression equation is the salary, and the independent variables are the experience and age of the employees. There are two ways to calculate the estimated coefficients b0 and b1: using the original sample observation and the deviation of the variables from their means. Now this definitely looks like a terrifying formula, but if you look closely the denominator is the same for both b1 and b2 and the numerator is a cross product of the 2 variables x1 and x2 along with y. Linear Regression. Just as simple linear regression defines a line in the (x,y) plane, the two variable multiple linear regression model Y = a + b1x1 + b2x2 + e is the equation of a plane in the (x1, x2, Y) space. Give a clap if you learnt something new today ! A is the intercept, b, c, and d are the slopes, and E is the residual value. P-values and coefficients in regression analysis work together to tell you which relationships in your model are statistically significant and the nature of those relationships. input[type=\'reset\'], However, I would also like to know whether the difference between the means of groups 2 and 3 is significant. Excel's data analysis toolpak can be used by users to perform data analysis and other important calculations. Calculate the values of the letters a, b1, b2. " /> Thank you! .entry-footer a.more-link{ Now, let us find out the relation between the salary of a group of employees in an organization, the number of years of experience, and the age of the employees. So, lets see in detail-What are Coefficients? Select the one with the lowest P-value. I have prepared a mini-research example of multiple linear regression analysis as exercise material. In the b0 = {} section of code, you call an intermediate result b, but later try to reference b1. color: #cd853f; input[type="submit"] The letter b is used to represent a sample estimate of a parameter. } 12. I'll try to give a more intuitive explanation first. A relatively simple form of the command (with labels and line plot) is Finally, I calculated y by y=b0 + b1*ln x1 + b2*ln x2 + b3*ln x3 +b4*ln x4 + b5*ln x5. Based on this background, the specifications of the multiple linear regression equation created by the researcher are as follows: b0, b1, b2 = regression estimation coefficient. (b) Write down the Regression equation of the problem |c) Calculate sales for 2010 if advertising were 14, 000 and . That is, given the presence of the other x-variables in the model, does a particular x-variable help us predict or explain the y-variable? { If you already know the summary statistics, you can calculate the equation of the regression line. Read More Lets look at the formulae: b1 = (x2_sq) (x1 y) ( x1 x2) (x2 y) / (x1_sq) (x2_sq) ( x1 x2)**2, b2 = (x1_sq) (x2 y) ( x1 x2) (x1 y) / (x1_sq) (x2_sq) ( x1 x2)**2. hr@degain.in This website uses cookies to improve your experience while you navigate through the website. var cli_flush_cache = true; } .go-to-top a:hover { .slider-buttons a:hover { #footer-navigation a:hover, For the audio-visual version, you can visit the KANDA DATA youtube channel. Based on the calculation results, the coefficient of determination value is 0.9285. Excepturi aliquam in iure, repellat, fugiat illum } .ld_button_640368d8e4edd.btn-icon-solid .btn-icon{background:rgb(247, 150, 34);}.ld_button_640368d8e4edd.btn-icon-circle.btn-icon-ripple .btn-icon:before{border-color:rgb(247, 150, 34);}.ld_button_640368d8e4edd{background-color:rgb(247, 150, 34);border-color:rgb(247, 150, 34);color:rgb(26, 52, 96);}.ld_button_640368d8e4edd .btn-gradient-border defs stop:first-child{stop-color:rgb(247, 150, 34);}.ld_button_640368d8e4edd .btn-gradient-border defs stop:last-child{stop-color:rgb(247, 150, 34);} .ai-viewport-3 { display: inherit !important;} }); .main-navigation li.menu-item-has-children > a:hover:after } font-style: italic; June 12, 2022 . Lets look at the formula for b0 first. B1 is the regression coefficient - how much we expect y to change as x increases. The regression formula for the above example will be y = MX + MX + b y= 604.17*-3.18+604.17*-4.06+0 y= -4377 background-color: rgba(220,101,67,0.5); Step 1: Calculate X12, X22, X1y, X2y and X1X2. If you want to write code to do regression (in which case saying "by hand" is super misleading), then you need a suitable computer -algorithm for solving X T X b = X T y -- the mathematically-obvious ways are dangerous. This website or its third-party tools use cookies, which are necessary to its functioning and required to achieve the purposes illustrated in the cookie policy. For this example, finding the solution is quite straightforward: b1 = 4.90 and b2 = 3.76. .go-to-top a:hover .fa-angle-up { right: 0; b1 value] keeping [other x variables i.e. .vivid, } .tag-links a, background: #cd853f; Likewise, bp is the difference in transportation costs between the current and previous years. Multiple Regression Calculator. a.sow-social-media-button:hover { background-color: #f1f1f1; .widget ul li a:hover, Finding the values of b0 and b1 that minimize this sum of squared errors gets us to the line of best fit. a { This website focuses on statistics, econometrics, data analysis, data interpretation, research methodology, and writing papers based on research. Say, we are predicting rent from square feet, and b1 say happens to be 2.5. The estimated linear regression equation is: = b 0 + b 1 *x 1 + b 2 *x 2. var Cli_Data = {"nn_cookie_ids":[],"cookielist":[]}; b1, b2, b3bn are coefficients for the independent variables x1, x2, x3, xn. #colophon .widget-title:after { } In this particular example, we will see which variable is the dependent variable and which variable is the independent variable. I have read the econometrics book by Koutsoyiannis (1977). Additional plots to consider are plots of residuals versus each. CFA And Chartered Financial Analyst Are Registered Trademarks Owned By CFA Institute. } If we start with a simple linear regression model with one predictor variable, \(x_1$$, then add a second predictor variable, $$x_2$$, $$SSE$$ will decrease (or stay the same) while $$SSTO$$ remains constant, and so $$R^2$$ will increase (or stay the same). Semi Circle Seekbar Android, Explanation of Regression Analysis Formula, Y= the dependent variable of the regression, X1=first independent variable of the regression, The x2=second independent variable of the regression, The x3=third independent variable of the regression. border: 1px solid #CD853F ; But, this doesn't necessarily mean that both $$x_1$$ and $$x_2$$ are not needed in a model with all the other predictors included. The bo (intercept) Coefficient can only be calculated if the coefficients b1 and b2 have been obtained. In Excel, researchers can create a table consisting of components for calculating b1, as shown in the image below: After creating a formula template in Excel, we need to calculate the average of the product sales variable (Y) and the advertising cost variable (X1). } .main-navigation a:hover, .main-navigation ul li.current-menu-item a, .main-navigation ul li.current_page_ancestor a, .main-navigation ul li.current-menu-ancestor a, .main-navigation ul li.current_page_item a, .main-navigation ul li:hover > a, .main-navigation ul li.current-menu-item.menu-item-has-children > a:after, .main-navigation li.menu-item-has-children > a:hover:after, .main-navigation li.page_item_has_children > a:hover:after {