

REVIEW ARTICLE 

Year : 2020  Volume
: 6
 Issue : 2  Page : 116122 

Ftest of overall significance in regression analysis simplified
Onchiri Sureiman^{1}, Callen Moraa Mangera^{2}
^{1} Department of Educational Planning and Management, Masinde Muliro University and Technology, Kakamega, Kenya ^{2} Department of Physiotherapy, The Nairobi Hospital, Nairobi, Kenya
Date of Submission  04Mar2020 
Date of Decision  30Apr2020 
Date of Acceptance  28Jun2020 
Date of Web Publication  27Aug2020 
Correspondence Address: Mr. Onchiri Sureiman Department of Educational Planning and Management, Masinde Muliro University and Technology, Kakamega Kenya
Source of Support: None, Conflict of Interest: None  Check 
DOI: 10.4103/jpcs.jpcs_18_20
Regression analysis is using the relationship between a known value and an unknown variable to estimate the unknown one. Here, an estimate of the dependent variable is made corresponding to given values of independent variables by placing the relationship between the variables in the form of a regression line. To determine how well the regression line obtained fits the given data points, Ftest of overall significance is conducted. The issues involved in the Ftest of overall significance are many and mathematics involved is rigorous, especially when more than two variables are involved. This study describes in details how the test can be conducted and finally gives the simplified approach of test using an online calculator.
Keywords: Ftest, hypothesis testing, online calculator, regression
How to cite this article: Sureiman O, Mangera CM. Ftest of overall significance in regression analysis simplified. J Pract Cardiovasc Sci 2020;6:11622 
How to cite this URL: Sureiman O, Mangera CM. Ftest of overall significance in regression analysis simplified. J Pract Cardiovasc Sci [serial online] 2020 [cited 2020 Sep 20];6:11622. Available from: http://www.jpcs.org/text.asp?2020/6/2/116/293527 
Introduction   
The term “regression” was first used in 1877 by Sir Francis Galton who made a study that showed that the height of children born to tall parents will tend to move back or “regress” toward the mean height of the population. He designated the word regression as name of the process of predicting one variable from another variable.^{[1]} Then came the term “multiple regression” to describe the process by which several variables are used to predict one another.^{[2]} The FTest of overall significance in regression is a test of whether or not your linear regression model provides a better fit to a dataset than a model with no predictor variables.
Assumptions Underlying FTest of Overall Significance in Regression Analysis   
The main assumptions include:
Linearity
Linear regression needs the relationship between the independent and dependent variables to be linear. It is also important to check for outliers since linear regression is sensitive to outlier effects. The linearity assumption can best be tested with scatter plots.^{[3]}
Normality
The linear regression analysis requires all variables to be multivariate normal.^{[4]} This assumption can best be checked with a histogram or a QQPlot. There are also a variety of statistical tests for normality, including the Kolmogorov–Smirnov test, the Shapiro–Wilk test, the Jarque–Bera test, and the Anderson–Darling test.^{[5]} When the data are not normally distributed a nonlinear transformation (e.g., logtransformation) might fix this issue.
Multicollinearity
Linear regression assumes that there is little or no multicollinearity in the data. Multicollinearity occurs when the independent variables are too highly correlated with each other.^{[3]} Multicollinearity may be tested with three central criteria:
 Tolerance – The tolerance measures the influence of one independent variable on all other independent variables; the tolerance is calculated with an initial linear regression analysis. Tolerance is defined as T = 1– R^{2} for these first step regression analysis.^{[6]} With T < 0.1 there might be multicollinearity in the data and with T < 0.01 there certainly is
 Correlation matrix – When computing the matrix of Pearson's Bivariate Correlation among all independent variables the correlation coefficients need to be smaller than 1^{[7]}
 Variance Inflation Factor (VIF) – The VIF of the linear regression is defined as VIF = 1/T. With VIF >5, there is an indication that multicollinearity may be present; with VIF >10, there is certainly multicollinearity among the variables.^{[3]} The simplest way to address the problem is to remove independent variables with high VIF values.
Homoscedasticity
The scatter plot is good way to check whether the data are homoscedastic (meaning the residuals are equal across the regression line). The Goldfeld–Quandt, Breush–Pagan, Park and White's tests can also be used to test for heteroscedasticity.^{[8]}
How to Interpret the FStatistic   
The Fstatistic is calculated as regression MS/residual MS. This statistic indicates whether the regression model provides a better fit to the data than a model that contains no independent variables. In essence, it tests if the regression model as a whole is useful. If the P < the significance level, there is sufficient evidence to conclude that the regression model fits the data better than the model with no predictor variables. This finding is good because it means that the predictor variables in the model actually improve the fit of the model. In general, if none of the predictor variables in the model are statistically significant, the overall F statistic is also not statistically significant.
Illustrative Examples on Determining FTest of Overall Significance in Regression Analysis   
This tutorial walks through examples of a regression analysis using two methods (manual and online calculator) providing an indepth explanation of how to read and interpret the output of a regression table.
Example 1
In estimating output (Y) of physiotherapist from a knowledge of his/her test score on the aptitude test (X_{1}) and years of experience (X_{2}) in a hospital, the [Table 1] summarizes the findings of the study.
H_{0}: Y = b_{0}
H_{1}: Y = b_{0 }+ b_{1}X_{1 }+ b_{2}X_{2}
Manual Computation of FTest of Overall Significance in Regression Analysis   
Obtaining the regression equation
The given data are reproduced in [Table 2]. [Table 2] also shows other inputs required for obtaining the regression equation.
The general form of multiple equation applicable in this case is:
Y = b_{0 }+ b_{1}X_{1 }+ b_{2}X_{2}
Moreover, the required normal equations to find the values of b_{0}, b_{1}, and b_{2} can be written as under:
Accordingly, the three equations are:
255 = 10 b_{0}+ 1354 b_{1}+ 53 b_{2}
37175 = 1354 b_{0}+ 194128 b_{1}+ 7374.5 b_{2}
1552 = 53 b_{0}+ 7347.5 b_{1}+ 363 b_{2}
Solving the three equations simultaneously, we obtain b_{0 }= −13.824567, b_{1}= 0.212167, and b_{2}= 1.999461. Thus, the regression equation of Y on X_{1} and X_{2} is: Y_{C}= 13.824567 + 0.212167 X_{2}+ 1.999461 X_{2.}
Calculation of R and Fratios
To determine the R and F statistic, we need to calculate total, explained and unexplained variation as shown in [Table 3].
Total variation (sum of squares total, SST) =974.5.
Explained variation (sum of square regression, SSR)=962 710
Unexplained variation (sum of squares error, SSE) =11.791
R square (R^{2}) =0.988, R= 0.984
Mean square regression (MS_{R})=481.355
Mean square error (MS_{E}) =1.684
F ==285 775
Goodness of fit
The F table value [Table 4] corresponding with degree of freedom n_{1}= 2 and n_{2}= 7 is 4.74. Since 285.775>4.74, we ignore the null hypothesis and conclude that Y = b_{0} or Y = b_{0}+ b_{1}X_{1}+ b_{2}X_{2}.
Validity checking
 Linearity: The relationship between the Y and X_{1} variables is linear [Figure 1] as well as the relationship between the Y and X_{2} variables [Figure 2]
 Normality
QQPlot illustrates [Figure 3] that all variables to be multivariate normal
 Multicollinearity
Tolerance = 1R^{2} = 10.987902 = 0.012098. With 0.012098= 0.01 but 0.012098 <0.1, there might be multicollinearity in the data.
 Homoscedasticityhomogeneity of variance.
The data are homoscedastic since the residuals are equal across the regression line [Figure 4] and [Figure 5].
Using an Online Linear Regression Analysis Calculator (Simplified Method)   
The Ftest of overall significance in regression analysis can be done through online calculators which are easily available in internet. For use friendly online calculator, you may visit this uniform locator http://www.statskingdom.com/410 multi_linear_regression.htm.
In the software, it is really easy to conduct an Ftest and most of the assumptions are preloaded. The calculator uses variables transformations, calculates the Linear equation, R, P value, outliers and the adjusted FisherPearson coefficient of skewness. After checking the residuals' normality, multicollinearity, homoscedasticity, and priori power, the program interprets the results. Then, it draws a histogram, a residuals QQplot, a correlation matrix, a residuals xplot and a distribution chart. You may transform the variables exclude any predictor or run backward stepwise selection automatically based on the predictor's P value.
The basic step for using and online calculator is to correctly fill in you data into it [Figure 6]. For instance, in the above example, we have to fill in the data in the columns of an online calculator. Click the calculate button.
Summary Output   
The output of the Ftest is summarized below by the regression equation, regression statistics [Figure 7], correlation matrix [Table 5], ANOVA [Table 6], coefficient table iteration I [Table 7], and residual graphs [Figure 8].
Regression equation is Y = −13.825 + 0.212 X_{1}+ 1.999 X_{2}
Validity Checking   
 Residual Normality: Linear regression assumes normality for residual errors. Shapiro–Wilk P = 0.664 [Figure 7]. It is assumed that the data are normally distributed
 HomoscedasticityHomogeneity of Variance: The White test P = 0.909 [Figure 7]. It is assumed that the variance is homogeneous
 MulticollinearityIntercorrelations among the Predictors: There is no multicollinearity concern as all the VIF values are smaller than 2.5 [Table 6] and [Table 7]
 Priori powerof the entire model (2 predictors): The priori power should be calculated before running the regression. Although the power is low: 0.134 [Figure 7], we reject the H_{0}.
Interpretation of the Output   
Y and X relationship
R square (R^{2}) equals 0.988. It means that the predictors (Xi) explain 98.8% of the variance of Y. Adjusted R square equals 0.984. The coefficient of multiple correlation® equals 0.994. It means that there is a very strong direct relationship between the predicted data (ŷ) and the observed data (y).
Goodness of fit
Righttailed F test is used to check if the entire regression model is statistically significant. From [Table 6], F (1, 7) = 285.802, P = 1.94764e7. Since P < α (0.05), we reject the H_{0}. The linear regression model, Y = b_{0 }+ b_{1}X_{1 }+ b_{2}X_{2}, provides a better fit than the model without the independent variables resulting in, Y = b_{0}.
As shown in [Table 5], P value for X_{1}= 6.59e7 and for X_{2}= 0.00000257. All the independent variables (Xi) are significant since P < α (0.05). The Yintercept (b): Twotailed, T = −7.701131, P = 0.000116139 [Table 7]. Hence, b is significantly different from zero.
Example 2
The data in [Table 8] are taken from a clinical trial to compare two hypotensive drugs used to lower the blood pressure during operations. The dependent variable, y, is the recovery time (in minutes) elapsing between the time at which the drug was discontinued and the time at which the systolic blood pressure had returned to 100 mmHg. The two predictors are quantity of drugs used in mg (x_{1}) and mean level of systolic blood pressure during hypotension in mmHg (x_{2}).
H_{0}: Y = b_{0}
H_{1}: Y = b_{0 }+ b_{1}X_{1 }+ b_{2}X_{2}
Using an Online Linear Regression Analysis Calculator (Simplified Method)   
To analyze the relationship between quantity of drugs used and mean level of systolic blood pressure during hypotension, we run a multiple linear regression using quantity of drugs used and mean level of systolic blood pressure during hypotension taken as the predictor variables and recovery time as the response variable. The output of the Ftest is summarized below by the regression equation, residual plots [Figure 9], correlation matrix [Table 9], ANOVA [Table 10] coefficient table iteration I [Table 11], and Regression statistics [Figure 10].
Regression equation is predicted Y = 58.603 + 53.688 X_{1}2.091 X_{2.}
Validity Checking   
 Residual Normality: Linear regression assumes normality for residual errors. Shapiro–Wilk P = 0.638 [Figure 10]. It is assumed that the data are normally distributed
 HomoscedasticityHomogeneity of Variance: The White test P value [Figure 10] equals 0.567 (F = 0.637). It is assumed that the variance is homogeneous
 ©MulticollinearityIntercorrelations among the Predictors. There is no multicollinearity concern as all the VIF values are smaller than 2.5 [Table 11]
 Priori powerof the Entire Model (2 Predictors): Although the power is low: 0.106 [Figure 10], we reject the H_{0.}
The power to prove each predictor significance is always lower than the power of the entire model.
Interpretation of the Output   
Y and X relationship
R square (R^{2}) equals 0.806. It means that the predictors (X_{i}) explain 80.6% of the variance of Y. Adjusted R square equals 0.728. The coefficient of multiple correlation ® equals 0.898. It means that there is a very strong direct relationship between the predicted data (ŷ) and the observed data (y).
Goodness of fit
Righttailed F test is used to check if the entire regression model is statistically significant. From [Table 10], F_{(1,}_{5)}= 10.382, P = 0.0166. Since P < α (0.05), we reject the H_{0}. The linear regression model, Y = b_{0 }+ b_{1}X_{1 }+ b_{2}X_{2,} provides a better fit than the model without the independent variables resulting in, Y = b_{0}.
As shown in [Table 11], P value for X_{1}= 0.0248 and for X_{2}= 0.0121. All the independent variables (Xi) are significant since P values< α (0.05). The Yintercept (b): Twotailed, T = 1.265, P = 0.262 [Table 11]. Hence, b is not significantly different from zero. It is still most likely recommended not to force b to be zero.
What Does an FTest of Overall Significance Test Tell and What it Does not   
The F statistic represents the ratio of the variance explained by the regression model (regression mean square) to the not explained variance (residuals mean square). It can be calculated easily using an online calculator in comparison to the manual approach. The Ftest of overall significance tests whether all of the predictor variables are jointly significant while the ttest of significance for each individual predictor variable merely tests whether each predictor variable is individually significant. Thus, the Ftest determines whether or not all of the predictor variables are jointly significant. It is possible that each predictor variable is not significant and yet the Ftest says that all of the predictor variables combined are jointly significant.
Financial support and sponsorship
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Conflicts of interest
There are no conflicts of interest.
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[Figure 1], [Figure 2], [Figure 3], [Figure 4], [Figure 5], [Figure 6], [Figure 7], [Figure 8], [Figure 9], [Figure 10]
[Table 1], [Table 2], [Table 3], [Table 4], [Table 5], [Table 6], [Table 7], [Table 8], [Table 9], [Table 10], [Table 11]
