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• Linear Regression in R | A Step-by-Step Guide & Examples

Linear Regression in R | A Step-by-Step Guide & Examples

Published on February 25, 2020 by Rebecca Bevans . Revised on May 10, 2024.

Linear regression is a regression model that uses a straight line to describe the relationship between variables . It finds the line of best fit through your data by searching for the value of the regression coefficient(s) that minimizes the total error of the model.

There are two main types of linear regression:

• Simple linear regression uses only one independent variable
• Multiple linear regression uses two or more independent variables

In this step-by-step guide, we will walk you through linear regression in R using two sample datasets.

Download the sample datasets to try it yourself.

Simple regression dataset Multiple regression dataset

Table of contents

Getting started in r, step 1: load the data into r, step 2: make sure your data meet the assumptions, step 3: perform the linear regression analysis, step 4: check for homoscedasticity, step 5: visualize the results with a graph, step 6: report your results, other interesting articles.

Start by downloading R and RStudio . Then open RStudio and click on File > New File > R Script .

As we go through each step , you can copy and paste the code from the text boxes directly into your script. To run the code, highlight the lines you want to run and click on the Run button on the top right of the text editor (or press ctrl + enter on the keyboard).

To install the packages you need for the analysis, run this code (you only need to do this once):

Next, load the packages into your R environment by running this code (you need to do this every time you restart R):

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Follow these four steps for each dataset:

• In RStudio, go to File > Import dataset  > From Text (base) .
• Choose the data file you have downloaded ( income.data or heart.data ), and an Import Dataset window pops up.
• In the Data Frame window, you should see an X (index) column and columns listing the data for each of the variables ( income and happiness or biking , smoking , and heart.disease ).
• Click on the Import button and the file should appear in your Environment tab on the upper right side of the RStudio screen.

After you’ve loaded the data, check that it has been read in correctly using summary() .

Simple regression

Because both our variables are quantitative , when we run this function we see a table in our console with a numeric summary of the data. This tells us the minimum, median , mean , and maximum values of the independent variable (income) and dependent variable (happiness):

Multiple regression

Again, because the variables are quantitative, running the code produces a numeric summary of the data for the independent variables (smoking and biking) and the dependent variable (heart disease):

We can use R to check that our data meet the four main assumptions for linear regression .

• Independence of observations (aka no autocorrelation)

Because we only have one independent variable and one dependent variable, we don’t need to test for any hidden relationships among variables.

If you know that you have autocorrelation within variables (i.e. multiple observations of the same test subject), then do not proceed with a simple linear regression! Use a structured model, like a linear mixed-effects model, instead.

To check whether the dependent variable follows a normal distribution , use the hist() function.

The observations are roughly bell-shaped (more observations in the middle of the distribution, fewer on the tails), so we can proceed with the linear regression.

The relationship between the independent and dependent variable must be linear. We can test this visually with a scatter plot to see if the distribution of data points could be described with a straight line.

The relationship looks roughly linear, so we can proceed with the linear model.

• Homoscedasticity  (aka homogeneity of variance )

This means that the prediction error doesn’t change significantly over the range of prediction of the model. We can test this assumption later, after fitting the linear model.

Use the cor() function to test the relationship between your independent variables and make sure they aren’t too highly correlated.

When we run this code, the output is 0.015. The correlation between biking and smoking is small (0.015 is only a 1.5% correlation), so we can include both parameters in our model.

Use the hist() function to test whether your dependent variable follows a normal distribution .

The distribution of observations is roughly bell-shaped, so we can proceed with the linear regression.

We can check this using two scatterplots: one for biking and heart disease, and one for smoking and heart disease.

Although the relationship between smoking and heart disease is a bit less clear, it still appears linear. We can proceed with linear regression.

• Homoscedasticity

We will check this after we make the model.

Now that you’ve determined your data meet the assumptions, you can perform a linear regression analysis to evaluate the relationship between the independent and dependent variables.

Simple regression: income and happiness

Let’s see if there’s a linear relationship between income and happiness in our survey of 500 people with incomes ranging from $15k to $75k, where happiness is measured on a scale of 1 to 10.

To perform a simple linear regression analysis and check the results, you need to run two lines of code. The first line of code makes the linear model, and the second line prints out the summary of the model:

The output looks like this:

This output table first presents the model equation, then summarizes the model residuals (see step 4).

The Coefficients section shows:

• The estimates ( Estimate ) for the model parameters – the value of the y-intercept (in this case 0.204) and the estimated effect of income on happiness (0.713).
• The standard error of the estimated values ( Std. Error ).
• The test statistic ( t value , in this case the t statistic ).
• The p value ( Pr(>| t | ) ), aka the probability of finding the given t statistic if the null hypothesis of no relationship were true.

The final three lines are model diagnostics – the most important thing to note is the p value (here it is 2.2e-16, or almost zero), which will indicate whether the model fits the data well.

From these results, we can say that there is a significant positive relationship between income and happiness ( p value < 0.001), with a 0.713-unit (+/- 0.01) increase in happiness for every unit increase in income.

Multiple regression: biking, smoking, and heart disease

Let’s see if there’s a linear relationship between biking to work, smoking, and heart disease in our imaginary survey of 500 towns. The rates of biking to work range between 1 and 75%, rates of smoking between 0.5 and 30%, and rates of heart disease between 0.5% and 20.5%.

To test the relationship, we first fit a linear model with heart disease as the dependent variable and biking and smoking as the independent variables. Run these two lines of code:

The estimated effect of biking on heart disease is -0.2, while the estimated effect of smoking is 0.178.

This means that for every 1% increase in biking to work, there is a correlated 0.2% decrease in the incidence of heart disease. Meanwhile, for every 1% increase in smoking, there is a 0.178% increase in the rate of heart disease.

The standard errors for these regression coefficients are very small, and the t statistics are very large (-147 and 50.4, respectively). The p values reflect these small errors and large t statistics. For both parameters, there is almost zero probability that this effect is due to chance.

Remember that these data are made up for this example, so in real life these relationships would not be nearly so clear!

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Before proceeding with data visualization, we should make sure that our models fit the homoscedasticity assumption of the linear model.

We can run plot(income.happiness.lm) to check whether the observed data meets our model assumptions:

Note that the par(mfrow()) command will divide the Plots window into the number of rows and columns specified in the brackets. So par(mfrow=c(2,2)) divides it up into two rows and two columns. To go back to plotting one graph in the entire window, set the parameters again and replace the (2,2) with (1,1).

These are the residual plots produced by the code:

Residuals are the unexplained variance . They are not exactly the same as model error, but they are calculated from it, so seeing a bias in the residuals would also indicate a bias in the error.

The most important thing to look for is that the red lines representing the mean of the residuals are all basically horizontal and centered around zero. This means there are no outliers or biases in the data that would make a linear regression invalid.

In the Normal Q-Qplot in the top right, we can see that the real residuals from our model form an almost perfectly one-to-one line with the theoretical residuals from a perfect model.

Based on these residuals, we can say that our model meets the assumption of homoscedasticity.

Again, we should check that our model is actually a good fit for the data, and that we don’t have large variation in the model error, by running this code:

As with our simple regression, the residuals show no bias, so we can say our model fits the assumption of homoscedasticity.

Next, we can plot the data and the regression line from our linear regression model so that the results can be shared.

Follow 4 steps to visualize the results of your simple linear regression.

• Plot the data points on a graph

• Add the linear regression line to the plotted data

Add the regression line using geom_smooth() and typing in lm as your method for creating the line. This will add the line of the linear regression as well as the standard error of the estimate (in this case +/- 0.01) as a light grey stripe surrounding the line:

• Add the equation for the regression line.

• Make the graph ready for publication

We can add some style parameters using theme_bw() and making custom labels using labs() .

This produces the finished graph that you can include in your papers:

The visualization step for multiple regression is more difficult than for simple regression, because we now have two predictors. One option is to plot a plane, but these are difficult to read and not often published.

We will try a different method: plotting the relationship between biking and heart disease at different levels of smoking. In this example, smoking will be treated as a factor with three levels, just for the purposes of displaying the relationships in our data.

There are 7 steps to follow.

• Create a new dataframe with the information needed to plot the model

Use the function expand.grid() to create a dataframe with the parameters you supply. Within this function we will:

• Create a sequence from the lowest to the highest value of your observed biking data;
• Choose the minimum, mean, and maximum values of smoking, in order to make 3 levels of smoking over which to predict rates of heart disease.

This will not create anything new in your console, but you should see a new data frame appear in the Environment tab. Click on it to view it.

• Predict the values of heart disease based on your linear model

Next we will save our ‘predicted y’ values as a new column in the dataset we just created.

• Round the smoking numbers to two decimals

This will make the legend easier to read later on.

• Change the ‘smoking’ variable into a factor

This allows us to plot the interaction between biking and heart disease at each of the three levels of smoking we chose.

• Plot the original data

• Add the regression lines

Because this graph has two regression coefficients, the stat_regline_equation() function won’t work here. But if we want to add our regression model to the graph, we can do so like this:

This is the finished graph that you can include in your papers!

In addition to the graph, include a brief statement explaining the results of the regression model.

If you want to know more about statistics , methodology , or research bias , make sure to check out some of our other articles with explanations and examples.

• Chi square test of independence
• Statistical power
• Descriptive statistics
• Degrees of freedom
• Pearson correlation
• Null hypothesis

Methodology

• Double-blind study
• Case-control study
• Research ethics
• Data collection
• Hypothesis testing
• Structured interviews

Research bias

• Hawthorne effect
• Unconscious bias
• Recall bias
• Halo effect
• Self-serving bias
• Information bias

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linearHypothesis: Test Linear Hypothesis

Description.

Generic function for testing a linear hypothesis, and methods for linear models, generalized linear models, multivariate linear models, linear and generalized linear mixed-effects models, generalized linear models fit with svyglm in the survey package, robust linear models fit with rlm in the MASS package, and other models that have methods for coef and vcov . For mixed-effects models, the tests are Wald chi-square tests for the fixed effects.

lht(model, ...)

# S3 method for default linearHypothesis(model, hypothesis.matrix, rhs=NULL, test=c("Chisq", "F"), vcov.=NULL, singular.ok=FALSE, verbose=FALSE, coef. = coef(model), suppress.vcov.msg=FALSE, error.df, ...)

# S3 method for lm linearHypothesis(model, hypothesis.matrix, rhs=NULL, test=c("F", "Chisq"), vcov.=NULL, white.adjust=c(FALSE, TRUE, "hc3", "hc0", "hc1", "hc2", "hc4"), singular.ok=FALSE, ...)

# S3 method for glm linearHypothesis(model, ...)

# S3 method for lmList linearHypothesis(model, ..., vcov.=vcov, coef.=coef)

# S3 method for nlsList linearHypothesis(model, ..., vcov.=vcov, coef.=coef)

# S3 method for mlm linearHypothesis(model, hypothesis.matrix, rhs=NULL, SSPE, V, test, idata, icontrasts=c("contr.sum", "contr.poly"), idesign, iterms, check.imatrix=TRUE, P=NULL, title="null hypothesis linear model r", singular.ok=FALSE, verbose=FALSE, ...) # S3 method for polr linearHypothesis(model, hypothesis.matrix, rhs=NULL, vcov., verbose=FALSE, ...) # S3 method for linearHypothesis.mlm print(x, SSP=TRUE, SSPE=SSP, digits=getOption("digits"), ...) # S3 method for lme linearHypothesis(model, hypothesis.matrix, rhs=NULL, vcov.=NULL, singular.ok=FALSE, verbose=FALSE, ...) # S3 method for mer linearHypothesis(model, hypothesis.matrix, rhs=NULL, vcov.=NULL, test=c("Chisq", "F"), singular.ok=FALSE, verbose=FALSE, ...) # S3 method for merMod linearHypothesis(model, hypothesis.matrix, rhs=NULL, vcov.=NULL, test=c("Chisq", "F"), singular.ok=FALSE, verbose=FALSE, ...) # S3 method for svyglm linearHypothesis(model, ...)

# S3 method for rlm linearHypothesis(model, ...)

# S3 method for survreg linearHypothesis(model, hypothesis.matrix, rhs=NULL, test=c("Chisq", "F"), vcov., verbose=FALSE, ...) matchCoefs(model, pattern, ...)

# S3 method for default matchCoefs(model, pattern, coef.=coef, ...)

# S3 method for lme matchCoefs(model, pattern, ...)

# S3 method for mer matchCoefs(model, pattern, ...)

# S3 method for merMod matchCoefs(model, pattern, ...)

# S3 method for mlm matchCoefs(model, pattern, ...)

# S3 method for lmList matchCoefs(model, pattern, ...)

For a univariate model, an object of class "anova"

which contains the residual degrees of freedom in the model, the difference in degrees of freedom, Wald statistic (either "F" or "Chisq" ), and corresponding p value. The value of the linear hypothesis and its covariance matrix are returned respectively as "value" and "vcov" attributes of the object (but not printed).

For a multivariate linear model, an object of class

"linearHypothesis.mlm" , which contains sums-of-squares-and-product matrices for the hypothesis and for error, degrees of freedom for the hypothesis and error, and some other information.

The returned object normally would be printed.

fitted model object. The default method of linearHypothesis works for models for which the estimated parameters can be retrieved by coef and the corresponding estimated covariance matrix by vcov . See the Details for more information.

matrix (or vector) giving linear combinations of coefficients by rows, or a character vector giving the hypothesis in symbolic form (see Details ).

right-hand-side vector for hypothesis, with as many entries as rows in the hypothesis matrix; can be omitted, in which case it defaults to a vector of zeroes. For a multivariate linear model, rhs is a matrix, defaulting to 0. This argument isn't available for F-tests for linear mixed models.

if FALSE (the default), a model with aliased coefficients produces an error; if TRUE , the aliased coefficients are ignored, and the hypothesis matrix should not have columns for them. For a multivariate linear model: will return the hypothesis and error SSP matrices even if the latter is singular; useful for computing univariate repeated-measures ANOVAs where there are fewer subjects than df for within-subject effects.

For the default linearHypothesis method, if an F-test is requested and if error.df is missing, the error degrees of freedom will be computed by applying the df.residual function to the model; if df.residual returns NULL or NA , then a chi-square test will be substituted for the F-test (with a message to that effect.

an optional data frame giving a factor or factors defining the intra-subject model for multivariate repeated-measures data. See Details for an explanation of the intra-subject design and for further explanation of the other arguments relating to intra-subject factors.

names of contrast-generating functions to be applied by default to factors and ordered factors, respectively, in the within-subject ``data''; the contrasts must produce an intra-subject model matrix in which different terms are orthogonal.

a one-sided model formula using the ``data'' in idata and specifying the intra-subject design.

the quoted name of a term, or a vector of quoted names of terms, in the intra-subject design to be tested.

check that columns of the intra-subject model matrix for different terms are mutually orthogonal (default, TRUE ). Set to FALSE only if you have already checked that the intra-subject model matrix is block-orthogonal.

transformation matrix to be applied to the repeated measures in multivariate repeated-measures data; if NULL and no intra-subject model is specified, no response-transformation is applied; if an intra-subject model is specified via the idata , idesign , and (optionally) icontrasts arguments, then P is generated automatically from the iterms argument.

in linearHypothesis method for mlm objects: optional error sum-of-squares-and-products matrix; if missing, it is computed from the model. In print method for linearHypothesis.mlm objects: if TRUE , print the sum-of-squares and cross-products matrix for error.

character string, "F" or "Chisq" , specifying whether to compute the finite-sample F statistic (with approximate F distribution) or the large-sample Chi-squared statistic (with asymptotic Chi-squared distribution). For a multivariate linear model, the multivariate test statistic to report --- one or more of "Pillai" , "Wilks" , "Hotelling-Lawley" , or "Roy" , with "Pillai" as the default.

an optional character string to label the output.

inverse of sum of squares and products of the model matrix; if missing it is computed from the model.

a function for estimating the covariance matrix of the regression coefficients, e.g., hccm , or an estimated covariance matrix for model . See also white.adjust . For the "lmList" and "nlsList" methods, vcov. must be a function (defaulting to vcov ) to be applied to each model in the list.

a vector of coefficient estimates. The default is to get the coefficient estimates from the model argument, but the user can input any vector of the correct length. For the "lmList" and "nlsList" methods, coef. must be a function (defaulting to coef ) to be applied to each model in the list.

logical or character. Convenience interface to hccm (instead of using the argument vcov. ). Can be set either to a character value specifying the type argument of hccm or TRUE , in which case "hc3" is used implicitly. The default is FALSE .

If TRUE , the hypothesis matrix, right-hand-side vector (or matrix), and estimated value of the hypothesis are printed to standard output; if FALSE (the default), the hypothesis is only printed in symbolic form and the value of the hypothesis is not printed.

an object produced by linearHypothesis.mlm .

if TRUE (the default), print the sum-of-squares and cross-products matrix for the hypothesis and the response-transformation matrix.

minimum number of signficiant digits to print.

a regular expression to be matched against coefficient names.

for internal use by methods that call the default method.

arguments to pass down.

Achim Zeileis and John Fox [email protected]

linearHypothesis computes either a finite-sample F statistic or asymptotic Chi-squared statistic for carrying out a Wald-test-based comparison between a model and a linearly restricted model. The default method will work with any model object for which the coefficient vector can be retrieved by coef and the coefficient-covariance matrix by vcov (otherwise the argument vcov. has to be set explicitly). For computing the F statistic (but not the Chi-squared statistic) a df.residual method needs to be available. If a formula method exists, it is used for pretty printing.

The method for "lm" objects calls the default method, but it changes the default test to "F" , supports the convenience argument white.adjust (for backwards compatibility), and enhances the output by the residual sums of squares. For "glm" objects just the default method is called (bypassing the "lm" method). The "svyglm" method also calls the default method.

Multinomial logit models fit by the multinom function in the nnet package invoke the default method, and the coefficient names are composed from the response-level names and conventional coefficient names, separated by a period ( "." ): see one of the examples below.

The function lht also dispatches to linearHypothesis .

The hypothesis matrix can be supplied as a numeric matrix (or vector), the rows of which specify linear combinations of the model coefficients, which are tested equal to the corresponding entries in the right-hand-side vector, which defaults to a vector of zeroes.

Alternatively, the hypothesis can be specified symbolically as a character vector with one or more elements, each of which gives either a linear combination of coefficients, or a linear equation in the coefficients (i.e., with both a left and right side separated by an equals sign). Components of a linear expression or linear equation can consist of numeric constants, or numeric constants multiplying coefficient names (in which case the number precedes the coefficient, and may be separated from it by spaces or an asterisk); constants of 1 or -1 may be omitted. Spaces are always optional. Components are separated by plus or minus signs. Newlines or tabs in hypotheses will be treated as spaces. See the examples below.

If the user sets the arguments coef. and vcov. , then the computations are done without reference to the model argument. This is like assuming that coef. is normally distibuted with estimated variance vcov. and the linearHypothesis will compute tests on the mean vector for coef. , without actually using the model argument.

A linear hypothesis for a multivariate linear model (i.e., an object of class "mlm" ) can optionally include an intra-subject transformation matrix for a repeated-measures design. If the intra-subject transformation is absent (the default), the multivariate test concerns all of the corresponding coefficients for the response variables. There are two ways to specify the transformation matrix for the repeated measures:

The transformation matrix can be specified directly via the P argument.

A data frame can be provided defining the repeated-measures factor or factors via idata , with default contrasts given by the icontrasts argument. An intra-subject model-matrix is generated from the one-sided formula specified by the idesign argument; columns of the model matrix corresponding to different terms in the intra-subject model must be orthogonal (as is insured by the default contrasts). Note that the contrasts given in icontrasts can be overridden by assigning specific contrasts to the factors in idata . The repeated-measures transformation matrix consists of the columns of the intra-subject model matrix corresponding to the term or terms in iterms . In most instances, this will be the simpler approach, and indeed, most tests of interests can be generated automatically via the Anova function.

matchCoefs is a convenience function that can sometimes help in formulating hypotheses; for example matchCoefs(mod, ":") will return the names of all interaction coefficients in the model mod .

Fox, J. (2016) Applied Regression Analysis and Generalized Linear Models , Third Edition. Sage.

Fox, J. and Weisberg, S. (2019) An R Companion to Applied Regression , Third Edition, Sage.

Hand, D. J., and Taylor, C. C. (1987) Multivariate Analysis of Variance and Repeated Measures: A Practical Approach for Behavioural Scientists. Chapman and Hall.

O'Brien, R. G., and Kaiser, M. K. (1985) MANOVA method for analyzing repeated measures designs: An extensive primer. Psychological Bulletin 97 , 316--333.

anova , Anova , waldtest , hccm , vcovHC , vcovHAC , coef , vcov

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Linear Regression

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Linear regression is used to predict the value of an outcome variable Y based on one or more input predictor variables X . The aim is to establish a linear relationship (a mathematical formula) between the predictor variable(s) and the response variable, so that, we can use this formula to estimate the value of the response Y , when only the predictors ( X s ) values are known.

Introduction

The aim of linear regression is to model a continuous variable Y as a mathematical function of one or more X variable(s), so that we can use this regression model to predict the Y when only the X is known. This mathematical equation can be generalized as follows:

Y  =  β 1  +  β 2 X  +  ϵ

where, β 1 is the intercept and β 2 is the slope. Collectively, they are called regression coefficients . ϵ is the error term, the part of Y the regression model is unable to explain.

Example Problem

For this analysis, we will use the cars dataset that comes with R by default. cars is a standard built-in dataset, that makes it convenient to demonstrate linear regression in a simple and easy to understand fashion. You can access this dataset simply by typing in cars in your R console. You will find that it consists of 50 observations(rows) and 2 variables (columns) – dist and speed . Lets print out the first six observations here..

Before we begin building the regression model, it is a good practice to analyze and understand the variables. The graphical analysis and correlation study below will help with this.

Graphical Analysis

The aim of this exercise is to build a simple regression model that we can use to predict Distance (dist) by establishing a statistically significant linear relationship with Speed (speed). But before jumping in to the syntax, lets try to understand these variables graphically. Typically, for each of the independent variables (predictors), the following plots are drawn to visualize the following behavior:

• Scatter plot : Visualize the linear relationship between the predictor and response
• Box plot : To spot any outlier observations in the variable. Having outliers in your predictor can drastically affect the predictions as they can easily affect the direction/slope of the line of best fit.
• Density plot : To see the distribution of the predictor variable. Ideally, a close to normal distribution (a bell shaped curve), without being skewed to the left or right is preferred. Let us see how to make each one of them.

Scatter Plot

Scatter plots can help visualize any linear relationships between the dependent (response) variable and independent (predictor) variables. Ideally, if you are having multiple predictor variables, a scatter plot is drawn for each one of them against the response, along with the line of best as seen below.

The scatter plot along with the smoothing line above suggests a linearly increasing relationship between the ‘dist’ and ‘speed’ variables. This is a good thing, because, one of the underlying assumptions in linear regression is that the relationship between the response and predictor variables is linear and additive.

BoxPlot – Check for outliers

Generally, any datapoint that lies outside the 1.5 * interquartile-range ( 1.5 *  I Q R ) is considered an outlier, where, IQR is calculated as the distance between the 25th percentile and 75th percentile values for that variable.

Density plot – Check if the response variable is close to normality

Correlation.

Correlation is a statistical measure that suggests the level of linear dependence between two variables, that occur in pair – just like what we have here in speed and dist. Correlation can take values between -1 to +1. If we observe for every instance where speed increases, the distance also increases along with it, then there is a high positive correlation between them and therefore the correlation between them will be closer to 1. The opposite is true for an inverse relationship, in which case, the correlation between the variables will be close to -1.

A value closer to 0 suggests a weak relationship between the variables. A low correlation (-0.2 < x < 0.2) probably suggests that much of variation of the response variable ( Y ) is unexplained by the predictor ( X ), in which case, we should probably look for better explanatory variables.

Build Linear Model

Now that we have seen the linear relationship pictorially in the scatter plot and by computing the correlation, lets see the syntax for building the linear model. The function used for building linear models is lm() . The lm() function takes in two main arguments, namely: 1. Formula 2. Data. The data is typically a data.frame and the formula is a object of class formula . But the most common convention is to write out the formula directly in place of the argument as written below.

Now that we have built the linear model, we also have established the relationship between the predictor and response in the form of a mathematical formula for Distance (dist) as a function for speed. For the above output, you can notice the ‘Coefficients’ part having two components: Intercept : -17.579, speed : 3.932 These are also called the beta coefficients. In other words, d i s t  =  I n t e r c e p t  + ( β  ∗  s p e e d ) => dist = −17.579 + 3.932∗speed

Linear Regression Diagnostics

Now the linear model is built and we have a formula that we can use to predict the dist value if a corresponding speed is known. Is this enough to actually use this model? NO! Before using a regression model, you have to ensure that it is statistically significant. How do you ensure this? Lets begin by printing the summary statistics for linearMod.

The p Value: Checking for statistical significance

The summary statistics above tells us a number of things. One of them is the model p-Value (bottom last line) and the p-Value of individual predictor variables (extreme right column under ‘Coefficients’). The p-Values are very important because, We can consider a linear model to be statistically significant only when both these p-Values are less that the pre-determined statistical significance level, which is ideally 0.05. This is visually interpreted by the significance stars at the end of the row. The more the stars beside the variable’s p-Value, the more significant the variable.

Null and alternate hypothesis

When there is a p-value, there is a hull and alternative hypothesis associated with it. In Linear Regression, the Null Hypothesis is that the coefficients associated with the variables is equal to zero. The alternate hypothesis is that the coefficients are not equal to zero (i.e. there exists a relationship between the independent variable in question and the dependent variable).

We can interpret the t-value something like this. A larger t-value indicates that it is less likely that the coefficient is not equal to zero purely by chance. So, higher the t-value, the better.

Pr(>|t|) or p-value is the probability that you get a t-value as high or higher than the observed value when the Null Hypothesis (the β coefficient is equal to zero or that there is no relationship) is true. So if the Pr(>|t|) is low, the coefficients are significant (significantly different from zero). If the Pr(>|t|) is high, the coefficients are not significant.

What this means to us? when p Value is less than significance level (< 0.05), we can safely reject the null hypothesis that the co-efficient β of the predictor is zero. In our case, linearMod , both these p-Values are well below the 0.05 threshold, so we can conclude our model is indeed statistically significant.

It is absolutely important for the model to be statistically significant before we can go ahead and use it to predict (or estimate) the dependent variable, otherwise, the confidence in predicted values from that model reduces and may be construed as an event of chance.

How to calculate the t Statistic and p-Values?

When the model co-efficients and standard error are known, the formula for calculating t Statistic and p-Value is as follows: $$t−Statistic = {β−coefficient \over Std.Error}$$

R-Squared and Adj R-Squared

The actual information in a data is the total variation it contains, remember?. What R-Squared tells us is the proportion of variation in the dependent (response) variable that has been explained by this model.

$$ R^{2} = 1 - \frac{SSE}{SST}$$

where, S S E is the sum of squared errors given by $SSE = \sum_{i}^{n} \left( y_{i} - \hat{y_{i}} \right) ^{2}$ and $SST = \sum_{i}^{n} \left( y_{i} - \bar{y_{i}} \right) ^{2}$ is the sum of squared total . Here, $\hat{y_{i}}$ is the fitted value for observation i and $\bar{y}$ is the mean of Y .

We don’t necessarily discard a model based on a low R-Squared value. Its a better practice to look at the AIC and prediction accuracy on validation sample when deciding on the efficacy of a model.

Now thats about R-Squared. What about adjusted R-Squared? As you add more X variables to your model, the R-Squared value of the new bigger model will always be greater than that of the smaller subset. This is because, since all the variables in the original model is also present, their contribution to explain the dependent variable will be present in the super-set as well, therefore, whatever new variable we add can only add (if not significantly) to the variation that was already explained. It is here, the adjusted R-Squared value comes to help. Adj R-Squared penalizes total value for the number of terms (read predictors) in your model. Therefore when comparing nested models, it is a good practice to look at adj-R-squared value over R-squared.

$$ R^{2}_{adj} = 1 - \frac{MSE}{MST}$$

where, M S E is the mean squared error given by $MSE = \frac{SSE}{\left( n-q \right)}$ and $MST = \frac{SST}{\left( n-1 \right)}$ is the mean squared total , where n is the number of observations and q is the number of coefficients in the model.

Therefore, by moving around the numerators and denominators, the relationship between R 2 and R a d j 2 becomes:

$$R^{2}_{adj} = 1 - \left( \frac{\left( 1 - R^{2}\right) \left(n-1\right)}{n-q}\right)$$

Standard Error and F-Statistic

Both standard errors and F-statistic are measures of goodness of fit.

$$Std. Error = \sqrt{MSE} = \sqrt{\frac{SSE}{n-q}}$$

$$F-statistic = \frac{MSR}{MSE}$$

where, n is the number of observations, q is the number of coefficients and M S R is the mean square regression , calculated as,

$$MSR=\frac{\sum_{i}^{n}\left( \hat{y_{i} - \bar{y}}\right)}{q-1} = \frac{SST - SSE}{q - 1}$$

AIC and BIC

The Akaike’s information criterion - AIC (Akaike, 1974) and the Bayesian information criterion - BIC (Schwarz, 1978) are measures of the goodness of fit of an estimated statistical model and can also be used for model selection. Both criteria depend on the maximized value of the likelihood function L for the estimated model.

The AIC is defined as:

A I C  = (−2) ×  l n ( L ) + (2× k )

where, k is the number of model parameters and the BIC is defined as:

B I C  = (−2) ×  l n ( L ) +  k  ×  l n ( n )

where, n is the sample size.

For model comparison, the model with the lowest AIC and BIC score is preferred.

How to know if the model is best fit for your data?

The most common metrics to look at while selecting the model are:

Predicting Linear Models

So far we have seen how to build a linear regression model using the whole dataset. If we build it that way, there is no way to tell how the model will perform with new data. So the preferred practice is to split your dataset into a 80:20 sample (training:test), then, build the model on the 80% sample and then use the model thus built to predict the dependent variable on test data.

Doing it this way, we will have the model predicted values for the 20% data (test) as well as the actuals (from the original dataset). By calculating accuracy measures (like min_max accuracy) and error rates (MAPE or MSE), we can find out the prediction accuracy of the model. Now, lets see how to actually do this..

Step 1: Create the training (development) and test (validation) data samples from original data.

Step 2: develop the model on the training data and use it to predict the distance on test data, step 3: review diagnostic measures..

From the model summary, the model p value and predictor’s p value are less than the significance level, so we know we have a statistically significant model. Also, the R-Sq and Adj R-Sq are comparative to the original model built on full data.

Step 4: Calculate prediction accuracy and error rates

A simple correlation between the actuals and predicted values can be used as a form of accuracy measure. A higher correlation accuracy implies that the actuals and predicted values have similar directional movement, i.e. when the actuals values increase the predicteds also increase and vice-versa.

Now lets calculate the Min Max accuracy and MAPE: $$MinMaxAccuracy = mean \left( \frac{min\left(actuals, predicteds\right)}{max\left(actuals, predicteds \right)} \right)$$

$$MeanAbsolutePercentageError \ (MAPE) = mean\left( \frac{abs\left(predicteds−actuals\right)}{actuals}\right)$$

k- Fold Cross validation

Suppose, the model predicts satisfactorily on the 20% split (test data), is that enough to believe that your model will perform equally well all the time? It is important to rigorously test the model’s performance as much as possible. One way is to ensure that the model equation you have will perform well, when it is ‘built’ on a different subset of training data and predicted on the remaining data.

How to do this is? Split your data into ‘k’ mutually exclusive random sample portions. Keeping each portion as test data, we build the model on the remaining (k-1 portion) data and calculate the mean squared error of the predictions. This is done for each of the ‘k’ random sample portions. Then finally, the average of these mean squared errors (for ‘k’ portions) is computed. We can use this metric to compare different linear models.

By doing this, we need to check two things:

• If the model’s prediction accuracy isn’t varying too much for any one particular sample, and
• If the lines of best fit don’t vary too much with respect the the slope and level.

In other words, they should be parallel and as close to each other as possible. You can find a more detailed explanation for interpreting the cross validation charts when you learn about advanced linear model building.

In the below plot, Are the dashed lines parallel? Are the small and big symbols are not over dispersed for one particular color?

Where to go from here?

We have covered the basic concepts about linear regression. Besides these, you need to understand that linear regression is based on certain underlying assumptions that must be taken care especially when working with multiple X s . Once you are familiar with that, the advanced regression models will show you around the various special cases where a different form of regression would be more suitable.

© 2016-17 Selva Prabhakaran. Powered by jekyll , knitr , and pandoc . This work is licensed under the Creative Commons License.

Step by Step I: Linear Models

Chapter 2 the null model, 2.1 objective.

This exercise uses a mock database for a Happiness Study with undergraduate and graduate students. In this exercise the research question is:

• How happy are the students?

2.2 Load the Data Base

The data base of the Happiness Study is available here . Download the data base load it in the R environment.

2.3 Model Estimation

2.3.1 data analysis equation.

The fundamental equation for data analysis is data = model + error . In mathematical notation:

\[Y_i=\bar{Y}_i + e_i\]

• \(Y_i\) is the value of the phenomena in subject i ;
• \(\bar{Y}_i\) an estimate of the value of the phenomena in subject i ;
• \(e_i\) is the diference between the value and the estimated value of the phenomena in subject i .

2.4 The Null Model

The null model is the simplest statistical model for a data set. It is given by a constant estimate:

\[\bar{Y}_i = b_0\]

• \(b_0\) corresponds to the best estimate for our data when there is no independent variable to be modeled.

In the case were the dependent variable is quantitative, \(b_0\) corresponds to the average of the dependent variable:

\[b_0=\bar{X}\]

• \(\bar{X}_i\) corresponds to the average of the dependent variable.

Represent the null model :

• Using the summary function and record the mean value;
• Plotting the average in a scatter with the functions plot and abline ;
• Create a constant in the database with the mean value and look at the data frame with the head function.

2.4.1 The Null Model Error

Models are estimates, simplifications of the data we observed. Consequently, every model has error. The error is given by the difference between the data and the model:

\[e_i=Y_i-\bar{Y}_i\]

Represent the null model error :

• Represent the difference between the data ( \(Y_i\) ) and the model ( \(\bar{Y}_i\) ) in the database.

The most intuitive solution to calculate the total error of a model is to add the error for each single case. However, this solution has a problem - the errors would cancel out because they have different signs.The statistical solution to calculate the total error of a model to perform the sum of square errors (SSE). Squaring the errors ensures that i) errors do not cancel out when added together and that ii) errors have weighted weights where larger errors weigh more than smaller errors (e.g., four errors of 1 squared and added together are worth 4 while that 3 errors of 0 and 1 error of 4 squared and added together are worth 16). The SSE is the conventional measure of error used in statistics to measure the performance of statistical models.

Represent the null model squared error :

• Represent the squared difference between the data ( \(Y_i\) ) and the model ( \(\bar{Y}_i\) ) in the database;
• Compute the sum of the errors and of the sum of the squared erros (SSE) using the sum function.

The SEQ indicates that the average happiness of 4.9495192 has an error of 84.922476 total square happiness. Note that the error in total square happiness is a problem because:

• it is a total (the more observations, the more error);
• it is squared (does not allow a direct comparison with the original measure).

Considering that the SSE is:

\[SSE=\displaystyle\sum\limits_{i=1}^n e_i^2\]

The average of the SSE is the variance and can be computed using:

\[s^2=\frac{SSE}{n-p}\]

The square root of the SSE is the standard-deviation and can be computed using:

\[s=\sqrt{s^2}\]

Compute the SSE, variance and standard-deviation using the formulas above and running the functions var and sd .

2.4.2 The Fundamental Principle of Data Analysis

The fundamental principle of data analysis is to minimize the error associated with a model. The lower the model error, the closer the model is to the data, and the better the model is. Error minimization is done using the Ordinary Squares Method (OLS). According to the OLS, the average is the best model to represent our data in a model without a VI and with a metric DV!

See bellow the SSE for possibles values of the null model . As expected, the model with lower SSE is the one with 4.9495192.

2.4.3 Model Estimation Conclusion

The null model and null model error show that:

• the mean happiness of the students is:
• the average error associated with the mean is:

2.5 The Experienced Researcher

Experienced Researchers report promptly the results of the model estimation, they don’t need to estimate in detail the null model and the error of the null mode. The calculation of the null model and error of the null model is a demonstration of the origin and meaning of these concepts (model and error) in the context of the simplest type of research problem possible.

• Why do we need models? Our data are usually bulky and of a form that is hard to communicate to others. The compact description provided by the models is much easier to communicate.
• How do we know which model is best? The minimization of error in linear models is done according to the Ordinary Least Squares Method (OLS). According to this method, when we have only one DV, the best null model is: for a metric DV, the average; for an ordinal RV, the median; and for a nominal DV, the mode.
• Why do we need a model as simple as the null model? The null model is the model against which we can compare the performance of more complex models that model the effects of IVs in a DV. A more complex model is only better than a null model if it manages to have less error than the null model!
• When do we look at models with VIs? The next classes are dedicated entirely to modeling the effects of IVs in a DV. For now, keep in mind the data analysis equation and the concepts of null model and null model error.
• Why don’t we talk about measures of central tendency or descriptive statistics? Averages, modes and medians are often referred to as measures of central tendency. Here we treat these measures as models, simplifications of our data.
• And why don’t we talk about dispersion measures? Sum of square errors, variance, standard deviation and standard error are often referred to as data dispersion measures. Here we treat these dispersion measures as error measures, the deviation between data and model.

2.6 Knowledge Assessment

Here is what you should know by now:

• What is the basic equation for data analysis?
• What is the null model?
• What are the most famous (null) models?
• What is the error associated with the null model?
• How is the error of a model measured?
• What is the fundamental principle of data analysis?

Hypothesis Tests in R

This tutorial covers basic hypothesis testing in R.

• Normality tests
• Shapiro-Wilk normality test
• Kolmogorov-Smirnov test
• Comparing central tendencies: Tests with continuous / discrete data
• One-sample t-test : Normally-distributed sample vs. expected mean
• Two-sample t-test : Two normally-distributed samples
• Wilcoxen rank sum : Two non-normally-distributed samples
• Weighted two-sample t-test : Two continuous samples with weights
• Comparing proportions: Tests with categorical data
• Chi-squared goodness of fit test : Sampled frequencies of categorical values vs. expected frequencies
• Chi-squared independence test : Two sampled frequencies of categorical values
• Weighted chi-squared independence test : Two weighted sampled frequencies of categorical values
• Comparing multiple groups: Tests with categorical and continuous / discrete data
• Analysis of Variation (ANOVA) : Normally-distributed samples in groups defined by categorical variable(s)
• Kruskal-Wallace One-Way Analysis of Variance : Nonparametric test of the significance of differences between two or more groups

Hypothesis Testing

Science is "knowledge or a system of knowledge covering general truths or the operation of general laws especially as obtained and tested through scientific method" (Merriam-Webster 2022) .

The idealized world of the scientific method is question-driven , with the collection and analysis of data determined by the formulation of research questions and the testing of hypotheses. Hypotheses are tentative assumptions about what the answers to your research questions may be.

• Formulate questions: How can I understand some phenomenon?
• Literature review: What does existing research say about my questions?
• Formulate hypotheses: What do I think the answers to my questions will be?
• Collect data: What data can I gather to test my hypothesis?
• Test hypotheses: Does the data support my hypothesis?
• Communicate results: Who else needs to know about this?
• Formulate questions: Frame missing knowledge about a phenomenon as research question(s).
• Literature review: A literature review is an investigation of what existing research says about the phenomenon you are studying. A thorough literature review is essential to identify gaps in existing knowledge you can fill, and to avoid unnecessarily duplicating existing research.
• Formulate hypotheses: Develop possible answers to your research questions.
• Collect data: Acquire data that supports or refutes the hypothesis.
• Test hypotheses: Run tools to determine if the data corroborates the hypothesis.
• Communicate results: Share your findings with the broader community that might find them useful.

While the process of knowledge production is, in practice, often more iterative than this waterfall model, the testing of hypotheses is usually a fundamental element of scientific endeavors involving quantitative data.

The Problem of Induction

The scientific method looks to the past or present to build a model that can be used to infer what will happen in the future. General knowledge asserts that given a particular set of conditions, a particular outcome will or is likely to occur.

The problem of induction is that we cannot be 100% certain that what we are assuming is a general principle is not, in fact, specific to the particular set of conditions when we made our empirical observations. We cannot prove that that such principles will hold true under future conditions or different locations that we have not yet experienced (Vickers 2014) .

The problem of induction is often associated with the 18th-century British philosopher David Hume . This problem is especially vexing in the study of human beings, where behaviors are a function of complex social interactions that vary over both space and time.

Falsification

One way of addressing the problem of induction was proposed by the 20th-century Viennese philosopher Karl Popper .

Rather than try to prove a hypothesis is true, which we cannot do because we cannot know all possible situations that will arise in the future, we should instead concentrate on falsification , where we try to find situations where a hypothesis is false. While you cannot prove your hypothesis will always be true, you only need to find one situation where the hypothesis is false to demonstrate that the hypothesis can be false (Popper 1962) .

If a hypothesis is not demonstrated to be false by a particular test, we have corroborated that hypothesis. While corroboration does not "prove" anything with 100% certainty, by subjecting a hypothesis to multiple tests that fail to demonstrate that it is false, we can have increasing confidence that our hypothesis reflects reality.

Null and Alternative Hypotheses

In scientific inquiry, we are often concerned with whether a factor we are considering (such as taking a specific drug) results in a specific effect (such as reduced recovery time).

To evaluate whether a factor results in an effect, we will perform an experiment and / or gather data. For example, in a clinical drug trial, half of the test subjects will be given the drug, and half will be given a placebo (something that appears to be the drug but is actually a neutral substance).

Because the data we gather will usually only be a portion (sample) of total possible people or places that could be affected (population), there is a possibility that the sample is unrepresentative of the population. We use a statistical test that considers that uncertainty when assessing whether an effect is associated with a factor.

• Statistical testing begins with an alternative hypothesis (H 1 ) that states that the factor we are considering results in a particular effect. The alternative hypothesis is based on the research question and the type of statistical test being used.
• Because of the problem of induction , we cannot prove our alternative hypothesis. However, under the concept of falsification , we can evaluate the data to see if there is a significant probability that our data falsifies our alternative hypothesis (Wilkinson 2012) .
• The null hypothesis (H 0 ) states that the factor has no effect. The null hypothesis is the opposite of the alternative hypothesis. The null hypothesis is what we are testing when we perform a hypothesis test.

The output of a statistical test like the t-test is a p -value. A p -value is the probability that any effects we see in the sampled data are the result of random sampling error (chance).

• If a p -value is greater than the significance level (0.05 for 5% significance) we fail to reject the null hypothesis since there is a significant possibility that our results falsify our alternative hypothesis.
• If a p -value is lower than the significance level (0.05 for 5% significance) we reject the null hypothesis and have corroborated (provided evidence for) our alternative hypothesis.

The calculation and interpretation of the p -value goes back to the central limit theorem , which states that random sampling error has a normal distribution.

Using our example of a clinical drug trial, if the mean recovery times for the two groups are close enough together that there is a significant possibility ( p > 0.05) that the recovery times are the same (falsification), we fail to reject the null hypothesis.

However, if the mean recovery times for the two groups are far enough apart that the probability they are the same is under the level of significance ( p < 0.05), we reject the null hypothesis and have corroborated our alternative hypothesis.

Significance means that an effect is "probably caused by something other than mere chance" (Merriam-Webster 2022) .

• The significance level (α) is the threshold for significance and, by convention, is usually 5%, 10%, or 1%, which corresponds to 95% confidence, 90% confidence, or 99% confidence, respectively.
• A factor is considered statistically significant if the probability that the effect we see in the data is a result of random sampling error (the p -value) is below the chosen significance level.
• A statistical test is used to evaluate whether a factor being considered is statistically significant (Gallo 2016) .

Type I vs. Type II Errors

Although we are making a binary choice between rejecting and failing to reject the null hypothesis, because we are using sampled data, there is always the possibility that the choice we have made is an error.

There are two types of errors that can occur in hypothesis testing.

• Type I error (false positive) occurs when a low p -value causes us to reject the null hypothesis, but the factor does not actually result in the effect.
• Type II error (false negative) occurs when a high p -value causes us to fail to reject the null hypothesis, but the factor does actually result in the effect.

The numbering of the errors reflects the predisposition of the scientific method to be fundamentally skeptical . Accepting a fact about the world as true when it is not true is considered worse than rejecting a fact about the world that actually is true.

Statistical Significance vs. Importance

When we fail to reject the null hypothesis, we have found information that is commonly called statistically significant . But there are multiple challenges with this terminology.

First, statistical significance is distinct from importance (NIST 2012) . For example, if sampled data reveals a statistically significant difference in cancer rates, that does not mean that the increased risk is important enough to justify expensive mitigation measures. All statistical results require critical interpretation within the context of the phenomenon being observed. People with different values and incentives can have different interpretations of whether statistically significant results are important.

Second, the use of 95% probability for defining confidence intervals is an arbitrary convention. This creates a good vs. bad binary that suggests a "finality and certitude that are rarely justified." Alternative approaches like Beyesian statistics that express results as probabilities can offer more nuanced ways of dealing with complexity and uncertainty (Clayton 2022) .

Science vs. Non-science

Not all ideas can be falsified, and Popper uses the distinction between falsifiable and non-falsifiable ideas to make a distinction between science and non-science. In order for an idea to be science it must be an idea that can be demonstrated to be false.

While Popper asserts there is still value in ideas that are not falsifiable, such ideas are not science in his conception of what science is. Such non-science ideas often involve questions of subjective values or unseen forces that are complex, amorphous, or difficult to objectively observe.

Example Data

As example data, this tutorial will use a table of anonymized individual responses from the CDC's Behavioral Risk Factor Surveillance System . The BRFSS is a "system of health-related telephone surveys that collect state data about U.S. residents regarding their health-related risk behaviors, chronic health conditions, and use of preventive services" (CDC 2019) .

A CSV file with the selected variables used in this tutorial is available here and can be imported into R with read.csv() .

Guidance on how to download and process this data directly from the CDC website is available here...

Variable Types

The publicly-available BRFSS data contains a wide variety of discrete, ordinal, and categorical variables. Variables often contain special codes for non-responsiveness or missing (NA) values. Examples of how to clean these variables are given here...

The BRFSS has a codebook that gives the survey questions associated with each variable, and the way that responses are encoded in the variable values.

Normality Tests

Tests are commonly divided into two groups depending on whether they are built on the assumption that the continuous variable has a normal distribution.

• Parametric tests presume a normal distribution.
• Non-parametric tests can work with normal and non-normal distributions.

The distinction between parametric and non-parametric techniques is especially important when working with small numbers of samples (less than 40 or so) from a larger population.

The normality tests given below do not work with large numbers of values, but with many statistical techniques, violations of normality assumptions do not cause major problems when large sample sizes are used. (Ghasemi and Sahediasi 2012) .

The Shapiro-Wilk Normality Test

• Data: A continuous or discrete sampled variable
• R Function: shapiro.test()
• Null hypothesis (H 0 ): The population distribution from which the sample is drawn is not normal
• History: Samuel Sanford Shapiro and Martin Wilk (1965)

This is an example with random values from a normal distribution.

This is an example with random values from a uniform (non-normal) distribution.

The Kolmogorov-Smirnov Test

The Kolmogorov-Smirnov is a more-generalized test than the Shapiro-Wilks test that can be used to test whether a sample is drawn from any type of distribution.

• Data: A continuous or discrete sampled variable and a reference probability distribution
• R Function: ks.test()
• Null hypothesis (H 0 ): The population distribution from which the sample is drawn does not match the reference distribution
• History: Andrey Kolmogorov (1933) and Nikolai Smirnov (1948)
• pearson.test() The Pearson Chi-square Normality Test from the nortest library. Lower p-values (closer to 0) means to reject the reject the null hypothesis that the distribution IS normal.

Modality Tests of Samples

Comparing two central tendencies: tests with continuous / discrete data, one sample t-test (two-sided).

The one-sample t-test tests the significance of the difference between the mean of a sample and an expected mean.

• Data: A continuous or discrete sampled variable and a single expected mean (μ)
• Parametric (normal distributions)
• R Function: t.test()
• Null hypothesis (H 0 ): The means of the sampled distribution matches the expected mean.
• History: William Sealy Gosset (1908)

t = ( Χ - μ) / (σ̂ / √ n )

• t : The value of t used to find the p-value
• Χ : The sample mean
• μ: The population mean
• σ̂: The estimate of the standard deviation of the population (usually the stdev of the sample
• n : The sample size

T-tests should only be used when the population is at least 20 times larger than its respective sample. If the sample size is too large, the low p-value makes the insignificant look significant. .

For example, we test a hypothesis that the mean weight in IL in 2020 is different than the 2005 continental mean weight.

Walpole et al. (2012) estimated that the average adult weight in North America in 2005 was 178 pounds. We could presume that Illinois is a comparatively normal North American state that would follow the trend of both increased age and increased weight (CDC 2021) .

The low p-value leads us to reject the null hypothesis and corroborate our alternative hypothesis that mean weight changed between 2005 and 2020 in Illinois.

One Sample T-Test (One-Sided)

Because we were expecting an increase, we can modify our hypothesis that the mean weight in 2020 is higher than the continental weight in 2005. We can perform a one-sided t-test using the alternative="greater" parameter.

The low p-value leads us to again reject the null hypothesis and corroborate our alternative hypothesis that mean weight in 2020 is higher than the continental weight in 2005.

Note that this does not clearly evaluate whether weight increased specifically in Illinois, or, if it did, whether that was caused by an aging population or decreasingly healthy diets. Hypotheses based on such questions would require more detailed analysis of individual data.

Although we can see that the mean cancer incidence rate is higher for counties near nuclear plants, there is the possiblity that the difference in means happened by accident and the nuclear plants have nothing to do with those higher rates.

The t-test allows us to test a hypothesis. Note that a t-test does not "prove" or "disprove" anything. It only gives the probability that the differences we see between two areas happened by chance. It also does not evaluate whether there are other problems with the data, such as a third variable, or inaccurate cancer incidence rate estimates.

Note that this does not prove that nuclear power plants present a higher cancer risk to their neighbors. It simply says that the slightly higher risk is probably not due to chance alone. But there are a wide variety of other other related or unrelated social, environmental, or economic factors that could contribute to this difference.

Box-and-Whisker Chart

One visualization commonly used when comparing distributions (collections of numbers) is a box-and-whisker chart. The boxes show the range of values in the middle 25% to 50% to 75% of the distribution and the whiskers show the extreme high and low values.

Although Google Sheets does not provide the capability to create box-and-whisker charts, Google Sheets does have candlestick charts , which are similar to box-and-whisker charts, and which are normally used to display the range of stock price changes over a period of time.

This video shows how to create a candlestick chart comparing the distributions of cancer incidence rates. The QUARTILE() function gets the values that divide the distribution into four equally-sized parts. This shows that while the range of incidence rates in the non-nuclear counties are wider, the bulk of the rates are below the rates in nuclear counties, giving a visual demonstration of the numeric output of our t-test.

While categorical data can often be reduced to dichotomous data and used with proportions tests or t-tests, there are situations where you are sampling data that falls into more than two categories and you would like to make hypothesis tests about those categories. This tutorial describes a group of tests that can be used with that type of data.

Two-Sample T-Test

When comparing means of values from two different groups in your sample, a two-sample t-test is in order.

The two-sample t-test tests the significance of the difference between the means of two different samples.

• Two normally-distributed, continuous or discrete sampled variables, OR
• A normally-distributed continuous or sampled variable and a parallel dichotomous variable indicating what group each of the values in the first variable belong to
• Null hypothesis (H 0 ): The means of the two sampled distributions are equal.

For example, given the low incomes and delicious foods prevalent in Mississippi, we might presume that average weight in Mississippi would be higher than in Illinois.

We test a hypothesis that the mean weight in IL in 2020 is less than the 2020 mean weight in Mississippi.

The low p-value leads us to reject the null hypothesis and corroborate our alternative hypothesis that mean weight in Illinois is less than in Mississippi.

While the difference in means is statistically significant, it is small (182 vs. 187), which should lead to caution in interpretation that you avoid using your analysis simply to reinforce unhelpful stigmatization.

Wilcoxen Rank Sum Test (Mann-Whitney U-Test)

The Wilcoxen rank sum test tests the significance of the difference between the means of two different samples. This is a non-parametric alternative to the t-test.

• Data: Two continuous sampled variables
• Non-parametric (normal or non-normal distributions)
• R Function: wilcox.test()
• Null hypothesis (H 0 ): For randomly selected values X and Y from two populations, the probability of X being greater than Y is equal to the probability of Y being greater than X.
• History: Frank Wilcoxon (1945) and Henry Mann and Donald Whitney (1947)

The test is is implemented with the wilcox.test() function.

• When the test is performed on one sample in comparison to an expected value around which the distribution is symmetrical (μ), the test is known as a Mann-Whitney U test .
• When the test is performed to compare two samples, the test is known as a Wilcoxon rank sum test .

For this example, we will use AVEDRNK3: During the past 30 days, on the days when you drank, about how many drinks did you drink on the average?

• 1 - 76: Number of drinks
• 77: Don’t know/Not sure
• 99: Refused
• NA: Not asked or Missing

The histogram clearly shows this to be a non-normal distribution.

Continuing the comparison of Illinois and Mississippi from above, we might presume that with all that warm weather and excellent food in Mississippi, they might be inclined to drink more. The means of average number of drinks per month seem to suggest that Mississippians do drink more than Illinoians.

We can test use wilcox.test() to test a hypothesis that the average amount of drinking in Illinois is different than in Mississippi. Like the t-test, the alternative can be specified as two-sided or one-sided, and for this example we will test whether the sampled Illinois value is indeed less than the Mississippi value.

The low p-value leads us to reject the null hypothesis and corroborates our hypothesis that average drinking is lower in Illinois than in Mississippi. As before, this tells us nothing about why this is the case.

Weighted Two-Sample T-Test

The downloadable BRFSS data is raw, anonymized survey data that is biased by uneven geographic coverage of survey administration (noncoverage) and lack of responsiveness from some segments of the population (nonresponse). The X_LLCPWT field (landline, cellphone weighting) is a weighting factor added by the CDC that can be assigned to each response to compensate for these biases.

The wtd.t.test() function from the weights library has a weights parameter that can be used to include a weighting factor as part of the t-test.

Comparing Proportions: Tests with Categorical Data

Chi-squared goodness of fit.

• Tests the significance of the difference between sampled frequencies of different values and expected frequencies of those values
• Data: A categorical sampled variable and a table of expected frequencies for each of the categories
• R Function: chisq.test()
• Null hypothesis (H 0 ): The relative proportions of categories in one variable are different from the expected proportions
• History: Karl Pearson (1900)
• Example Question: Are the voting preferences of voters in my district significantly different from the current national polls?

For example, we test a hypothesis that smoking rates changed between 2000 and 2020.

In 2000, the estimated rate of adult smoking in Illinois was 22.3% (Illinois Department of Public Health 2004) .

The variable we will use is SMOKDAY2: Do you now smoke cigarettes every day, some days, or not at all?

• 1: Current smoker - now smokes every day
• 2: Current smoker - now smokes some days
• 3: Not at all
• 7: Don't know
• NA: Not asked or missing - NA is used for people who have never smoked

We subset only yes/no responses in Illinois and convert into a dummy variable (yes = 1, no = 0).

The listing of the table as percentages indicates that smoking rates were halved between 2000 and 2020, but since this is sampled data, we need to run a chi-squared test to make sure the difference can't be explained by the randomness of sampling.

In this case, the very low p-value leads us to reject the null hypothesis and corroborates the alternative hypothesis that smoking rates changed between 2000 and 2020.

Chi-Squared Contingency Analysis / Test of Independence

• Tests the significance of the difference between frequencies between two different groups
• Data: Two categorical sampled variables
• Null hypothesis (H 0 ): The relative proportions of one variable are independent of the second variable.

We can also compare categorical proportions between two sets of sampled categorical variables.

The chi-squared test can is used to determine if two categorical variables are independent. What is passed as the parameter is a contingency table created with the table() function that cross-classifies the number of rows that are in the categories specified by the two categorical variables.

The null hypothesis with this test is that the two categories are independent. The alternative hypothesis is that there is some dependency between the two categories.

For this example, we can compare the three categories of smokers (daily = 1, occasionally = 2, never = 3) across the two categories of states (Illinois and Mississippi).

The low p-value leads us to reject the null hypotheses that the categories are independent and corroborates our hypotheses that smoking behaviors in the two states are indeed different.

p-value = 1.516e-09

Weighted Chi-Squared Contingency Analysis

As with the weighted t-test above, the weights library contains the wtd.chi.sq() function for incorporating weighting into chi-squared contingency analysis.

As above, the even lower p-value leads us to again reject the null hypothesis that smoking behaviors are independent in the two states.

Suppose that the Macrander campaign would like to know how partisan this election is. If people are largely choosing to vote along party lines, the campaign will seek to get their base voters out to the polls. If people are splitting their ticket, the campaign may focus their efforts more broadly.

In the example below, the Macrander campaign took a small poll of 30 people asking who they wished to vote for AND what party they most strongly affiliate with.

The output of table() shows fairly strong relationship between party affiliation and candidates. Democrats tend to vote for Macrander, while Republicans tend to vote for Stewart, while independents all vote for Miller.

This is reflected in the very low p-value from the chi-squared test. This indicates that there is a very low probability that the two categories are independent. Therefore we reject the null hypothesis.

In contrast, suppose that the poll results had showed there were a number of people crossing party lines to vote for candidates outside their party. The simulated data below uses the runif() function to randomly choose 50 party names.

The contingency table() shows no clear relationship between party affiliation and candidate. This is validated quantitatively by the chi-squared test. The fairly high p-value of 0.4018 indicates a 40% chance that the two categories are independent. Therefore, we fail to reject the null hypothesis and the campaign should focus their efforts on the broader electorate.

The warning message given by the chisq.test() function indicates that the sample size is too small to make an accurate analysis. The simulate.p.value = T parameter adds Monte Carlo simulation to the test to improve the estimation and get rid of the warning message. However, the best way to get rid of this message is to get a larger sample.

Comparing Categorical and Continuous Variables

Analysis of variation (anova).

Analysis of Variance (ANOVA) is a test that you can use when you have a categorical variable and a continuous variable. It is a test that considers variability between means for different categories as well as the variability of observations within groups.

There are a wide variety of different extensions of ANOVA that deal with covariance (ANCOVA), multiple variables (MANOVA), and both of those together (MANCOVA). These techniques can become quite complicated and also assume that the values in the continuous variables have a normal distribution.

• Data: One or more categorical (independent) variables and one continuous (dependent) sampled variable
• R Function: aov()
• Null hypothesis (H 0 ): There is no difference in means of the groups defined by each level of the categorical (independent) variable
• History: Ronald Fisher (1921)
• Example Question: Do low-, middle- and high-income people vary in the amount of time they spend watching TV?

As an example, we look at the continuous weight variable (WEIGHT2) split into groups by the eight income categories in INCOME2: Is your annual household income from all sources?

• 1: Less than $10,000
• 2: $10,000 to less than $15,000
• 3: $15,000 to less than $20,000
• 4: $20,000 to less than $25,000
• 5: $25,000 to less than $35,000
• 6: $35,000 to less than $50,000
• 7: $50,000 to less than $75,000)
• 8: $75,000 or more

The barplot() of means does show variation among groups, although there is no clear linear relationship between income and weight.

To test whether this variation could be explained by randomness in the sample, we run the ANOVA test.

The low p-value leads us to reject the null hypothesis that there is no difference in the means of the different groups, and corroborates the alternative hypothesis that mean weights differ based on income group.

However, it gives us no clear model for describing that relationship and offers no insights into why income would affect weight, especially in such a nonlinear manner.

Suppose you are performing research into obesity in your city. You take a sample of 30 people in three different neighborhoods (90 people total), collecting information on health and lifestyle. Two variables you collect are height and weight so you can calculate body mass index . Although this index can be misleading for some populations (notably very athletic people), ordinary sedentary people can be classified according to BMI:

Average BMI in the US from 2007-2010 was around 28.6 and rising, standard deviation of around 5 .

You would like to know if there is a difference in BMI between different neighborhoods so you can know whether to target specific neighborhoods or make broader city-wide efforts. Since you have more than two groups, you cannot use a t-test().

Kruskal-Wallace One-Way Analysis of Variance

A somewhat simpler test is the Kruskal-Wallace test which is a nonparametric analogue to ANOVA for testing the significance of differences between two or more groups.

• R Function: kruskal.test()
• Null hypothesis (H 0 ): The samples come from the same distribution.
• History: William Kruskal and W. Allen Wallis (1952)

For this example, we will investigate whether mean weight varies between the three major US urban states: New York, Illinois, and California.

To test whether this variation could be explained by randomness in the sample, we run the Kruskal-Wallace test.

The low p-value leads us to reject the null hypothesis that the samples come from the same distribution. This corroborates the alternative hypothesis that mean weights differ based on state.

A convienent way of visualizing a comparison between continuous and categorical data is with a box plot , which shows the distribution of a continuous variable across different groups:

A percentile is the level at which a given percentage of the values in the distribution are below: the 5th percentile means that five percent of the numbers are below that value.

The quartiles divide the distribution into four parts. 25% of the numbers are below the first quartile. 75% are below the third quartile. 50% are below the second quartile, making it the median.

Box plots can be used with both sampled data and population data.

The first parameter to the box plot is a formula: the continuous variable as a function of (the tilde) the second variable. A data= parameter can be added if you are using variables in a data frame.

The chi-squared test can be used to determine if two categorical variables are independent of each other.

Life With Data

• by bprasad26

How to Use the linearHypothesis() Function in R

The linearHypothesis() function is a valuable statistical tool in R programming. It’s provided in the car package and is used to perform hypothesis testing for a linear model’s coefficients.

To fully grasp the utility of linearHypothesis() , we must understand the basic principles of linear regression and hypothesis testing in the context of model fitting.

Understanding Hypothesis Testing in Regression Analysis

In regression analysis, it’s common to perform hypothesis tests on the model’s coefficients to determine whether the predictors are statistically significant. The null hypothesis asserts that the predictor has no effect on the outcome variable, i.e., its coefficient equals zero. Rejecting the null hypothesis (based on a small p-value, usually less than 0.05) suggests that there’s a statistically significant relationship between the predictor and the outcome variable.

The linearHypothesis( ) Function

linearHypothesis() is a function in R that tests the general linear hypothesis for a model object for which a formula method exists, using a specified test statistic. It allows the user to define a broader set of null hypotheses than just assuming individual coefficients equal to zero.

The linearHypothesis() function can be especially useful for comparing nested models or testing whether a group of variables significantly contributes to the model.

Here’s the basic usage of linearHypothesis() :

In this function:

• model is the model object for which the linear hypothesis is to be tested.
• hypothesis.matrix specifies the null hypotheses.
• rhs is the right-hand side of the linear hypotheses; typically set to 0.
• ... are additional arguments, such as the test argument to specify the type of test statistic to be used (“F” for F-test, “Chisq” for chi-squared test, etc.).

Installing and Loading the Required Package

linearHypothesis() is part of the car package. If you haven’t installed this package yet, you can do so using the following command:

Once installed, load it into your R environment with the library() function:

Using linearHypothesis( ) in Practice

Let’s demonstrate the use of linearHypothesis() with a practical example. We’ll use the mtcars dataset that’s built into R. This dataset comprises various car attributes, and we’ll model miles per gallon (mpg) based on horsepower (hp), weight (wt), and the number of cylinders (cyl).

We first fit a linear model using the lm() function:

Let’s say we want to test the hypothesis that the coefficients for hp and wt are equal to zero. We can set up this hypothesis test using linearHypothesis() :

This command will output the Residual Sum of Squares (RSS) for the model under the null hypothesis, the RSS for the full model, the test statistic, and the p-value for the test. A low p-value suggests that we should reject the null hypothesis.

Using linearHypothesis( ) for Testing Nested Models

linearHypothesis() can also be useful for testing nested models, i.e., comparing a simpler model to a more complex one where the simpler model is a special case of the complex one.

For instance, suppose we want to test if both hp and wt can be dropped from our model without a significant loss of fit. We can formulate this as the null hypothesis that the coefficients for hp and wt are simultaneously zero:

This gives a p-value for the F-test of the hypothesis that these coefficients are zero. If the p-value is small, we reject the null hypothesis and conclude that dropping these predictors from the model would significantly degrade the model fit.

Limitations and Considerations

The linearHypothesis() function is a powerful tool for hypothesis testing in the context of model fitting. However, it’s important to consider the limitations and assumptions of this function. The linearHypothesis() function assumes that the errors of the model are normally distributed and have equal variance. Violations of these assumptions can lead to incorrect results.

As with any statistical function, it’s crucial to have a good understanding of your data and the theory behind the statistical methods you’re using.

The linearHypothesis() function in R is a powerful tool for testing linear hypotheses about a model’s coefficients. This function is very flexible and can be used in various scenarios, including testing the significance of individual predictors and comparing nested models.

Understanding and properly using linearHypothesis() can enhance your data analysis capabilities and help you extract meaningful insights from your data.

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An R Introduction to Statistics

• Terms of Use
• Significance Test for Linear Regression

Assume that the error term ϵ in the linear regression model is independent of x , and is normally distributed , with zero mean and constant variance . We can decide whether there is any significant relationship between x and y by testing the null hypothesis that β = 0 .

Decide whether there is a significant relationship between the variables in the linear regression model of the data set faithful at .05 significance level.

We apply the lm function to a formula that describes the variable eruptions by the variable waiting , and save the linear regression model in a new variable eruption.lm .

Then we print out the F-statistics of the significance test with the summary function.

As the p-value is much less than 0.05, we reject the null hypothesis that β = 0 . Hence there is a significant relationship between the variables in the linear regression model of the data set faithful .

Further detail of the summary function for linear regression model can be found in the R documentation.

• Elementary Statistics with R
• linear regression
• significance test

R Tutorial eBook

R Tutorials

• Combining Vectors
• Vector Arithmetics
• Vector Index
• Numeric Index Vector
• Logical Index Vector
• Named Vector Members
• Matrix Construction
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• Data Frame Column Vector
• Data Frame Column Slice
• Data Frame Row Slice
• Data Import
• Frequency Distribution of Qualitative Data
• Relative Frequency Distribution of Qualitative Data
• Category Statistics
• Frequency Distribution of Quantitative Data
• Relative Frequency Distribution of Quantitative Data
• Cumulative Frequency Distribution
• Cumulative Frequency Graph
• Cumulative Relative Frequency Distribution
• Cumulative Relative Frequency Graph
• Stem-and-Leaf Plot
• Scatter Plot
• Interquartile Range
• Standard Deviation
• Correlation Coefficient
• Central Moment
• Binomial Distribution
• Poisson Distribution
• Continuous Uniform Distribution
• Exponential Distribution
• Normal Distribution
• Chi-squared Distribution
• Student t Distribution
• F Distribution
• Point Estimate of Population Mean
• Interval Estimate of Population Mean with Known Variance
• Interval Estimate of Population Mean with Unknown Variance
• Sampling Size of Population Mean
• Point Estimate of Population Proportion
• Interval Estimate of Population Proportion
• Sampling Size of Population Proportion
• Lower Tail Test of Population Mean with Known Variance
• Upper Tail Test of Population Mean with Known Variance
• Two-Tailed Test of Population Mean with Known Variance
• Lower Tail Test of Population Mean with Unknown Variance
• Upper Tail Test of Population Mean with Unknown Variance
• Two-Tailed Test of Population Mean with Unknown Variance
• Lower Tail Test of Population Proportion
• Upper Tail Test of Population Proportion
• Two-Tailed Test of Population Proportion
• Type II Error in Lower Tail Test of Population Mean with Known Variance
• Type II Error in Upper Tail Test of Population Mean with Known Variance
• Type II Error in Two-Tailed Test of Population Mean with Known Variance
• Type II Error in Lower Tail Test of Population Mean with Unknown Variance
• Type II Error in Upper Tail Test of Population Mean with Unknown Variance
• Type II Error in Two-Tailed Test of Population Mean with Unknown Variance
• Population Mean Between Two Matched Samples
• Population Mean Between Two Independent Samples
• Comparison of Two Population Proportions
• Multinomial Goodness of Fit
• Chi-squared Test of Independence
• Completely Randomized Design
• Randomized Block Design
• Factorial Design
• Wilcoxon Signed-Rank Test
• Mann-Whitney-Wilcoxon Test
• Kruskal-Wallis Test
• Estimated Simple Regression Equation
• Coefficient of Determination
• Confidence Interval for Linear Regression
• Prediction Interval for Linear Regression
• Residual Plot
• Standardized Residual
• Normal Probability Plot of Residuals
• Estimated Multiple Regression Equation
• Multiple Coefficient of Determination
• Adjusted Coefficient of Determination
• Significance Test for MLR
• Confidence Interval for MLR
• Prediction Interval for MLR
• Estimated Logistic Regression Equation
• Significance Test for Logistic Regression
• Distance Matrix by GPU
• Hierarchical Cluster Analysis
• Kendall Rank Coefficient
• Significance Test for Kendall's Tau-b
• Support Vector Machine with GPU
• Support Vector Machine with GPU, Part II
• Bayesian Classification with Gaussian Process
• Hierarchical Linear Model
• Installing GPU Packages

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How to interpret the results of linearHypothesis function when comparing regression coefficients?

I used linearHypothesis function in order to test whether two regression coefficients are significantly different. Do you have any idea how to interpret these results?

Here is my output:

• 1 Pr(>F) is the p-value of the test, and this is the output of interest. You want the interpretation of every output ? –  Stéphane Laurent Commented Feb 11, 2019 at 12:40

3 Answers 3

Short Answer

Your F statistic is 104.34 and its p-value 2.2e-16. The corresponding p-value suggests that we can reject the null hypothesis that both coefficients cancel each other at any level of significance commonly used in practice.

Were your p-value greater than 0.05, it is accustomed that you would not reject the null hypothesis.

Long Answer

The linearHypothesis function tests whether the difference between the coefficients is significant. In your example, whether the two betas cancel each other out β1 − β2 = 0.

Linear hypothesis tests are performed using F-statistics. They compare your estimated model against a restrictive model which requires your hypothesis (restriction) to be true.

An alternative linear hypothesis testing would be to test whether β1 or β2 are nonzero, so we jointly test the hypothesis β1=0 and β2 = 0 rather than testing each one at a time. Here the null is rejected when one is rejected. Rejection here means that at least one of your hypotheses can be rejected. In other words provide both linear restrictions to be tested as strings

Here are few examples of the multitude of ways you can test hypothese:

You can test a linear combination of coeffecients

joint probability

Aside from the t statistics, which test for the predictive power of each variable in the presence of all the others, another test which can be used is the F-test. (this is the F-test that you would get at the bottom of a linear model)

This tests the null hypothesis that all of the β’s are equal to zero against the alternative that allows them to take any values. If we reject this null hypothesis (which we do because the p-value is small), then this is the same as saying there is enough evidence to conclude that at least one of the covariates has predictive power in our linear model, i.e. that using a regression is predictively ‘better’ than just guessing the average.

So basically, you are testing whether all coefficients are different from zero or some other arbitrary linear hypothesis, as opposed to a t-test where you are testing individual coefficients.

The answer given above is detailed enough except that for this test we are more interested in the two variables hence the linear hypothesis does not investigate the null hypothesis that all of the β’s are equal to zero against the alternative that allows them to take any values but just for two variables of interest which makes this test equivalent to a t-test.

• 1 As it’s currently written, your answer is unclear. Please edit to add additional details that will help others understand how this addresses the question asked. You can find more information on how to write good answers in the help center . –  Community Bot Commented Sep 14, 2022 at 19:22

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binomial glm with null-hypothesis = 0.33

I have a dataset and need to test if the values are statistical different to 0.33 (=1/3). Flies have the choice between 3 types of berries. So, if they do not care about the type they would lay 1/3 of their eggs in each typ of berry and that would be the null-hypothesis. I want to know, if they layed signifikcant more than 1/3 of their eggs in one specific type of berry.

With this model I would test if the values are different from 0.5:

But I need a test against 0.33. As far as I understand the answers to other questions regarding this topic I can change the test by including an offset command, but I quit do not understand what exactly to include to test against 0.33.

Many thanks for any help you can provide.

• hypothesis-testing
• generalized-linear-model

• 2 $\begingroup$ Could you clarify why you are using a quasibinomial glm for what looks like a straight test of proportions? Why not binom.test or chisq.test ? If you do particularly want to use a glm for whatever reason, why a quasi model? $\endgroup$ –  Glen_b Commented Oct 21, 2018 at 7:11
• $\begingroup$ Sorry, that is the model I am using so far to test binomial values. I learned that I have to use a quasi model, if the diepersal is not within 0.5 and 1. I have to admit, aswering your question further is far beyond my state of knowledge. $\endgroup$ –  R. Kienzle Commented Oct 22, 2018 at 13:31

3 Answers 3

Edit: This answer addresses the simple case in which there are three counts and the distribution of these counts are tested against a null hypothesis of equal distribution. Comments by the OP suggest the set-up of the experiment is more complicated than this.

Aside from the multinomial test mentioned by @a_statistician , you could also use a chi-square goodness-of-fit test. This test is probably more common, but also may be inappropriate when there are low expected counts.

Probably more beneficial would be looking at the multinomial confidence intervals for the proportions.

Plotting these gives us a basis for comparison.

(Caveat: I am the author of these pages.)

http://rcompanion.org/handbook/H_03.html

http://rcompanion.org/handbook/H_02.html

• $\begingroup$ With the chi-square goodness of fit test for multinomial distribution, I again get a simular error message as for the goodness of fit test for multinomial distribution, that x and p have to have the same number of elements. The model for looking at the multinomial confidence intervals for the proportions, is runing but the plot looks very weird and I have to admit, that this suggestion might be beyond my understanding. $\endgroup$ –  R. Kienzle Commented Oct 22, 2018 at 13:58
• $\begingroup$ Yes, x and p need to have the same number of elements, which is three in your case. Each element in x should be a single number, the count for that berry. $\endgroup$ –  Sal Mangiafico Commented Oct 22, 2018 at 14:13
• $\begingroup$ Ok and how do I include, that I have several replicates? Or do you suggest just testing the overall sum of eggs per berry within the 10 replicates? $\endgroup$ –  R. Kienzle Commented Oct 22, 2018 at 14:51
• $\begingroup$ In that case I would recommend a generalized linear model for count data, like Poisson or negative binomial, where each replicate is an observation. But it might make sense to sum them up into three counts. $\endgroup$ –  Sal Mangiafico Commented Oct 22, 2018 at 15:30
• $\begingroup$ The problem with a poisson model is, that I have in addition 4 different treatments with 1,3, 6 and 10 berries of each type. Since the flies tend to lay more eggs if they have accsses to more berries I would have to work with eggs per berry by dividing the sums through 3,6 or 10. Therefore I get data with not whole numbers and the poission model will not run. That is why I wanted to go with binomial. $\endgroup$ –  R. Kienzle Commented Oct 22, 2018 at 15:46

If the probability of the event was 0.33, then the log-odds of the event (the linear predictor for a binomial and quasibinomial model) is:

$$\mbox{logit} 0.33 = -0.7081851$$

To fit the model in R:

The statistical significance of the intercept term will tell you whether the risk is different from that value.

The quasibinomial works in spite of the constraint of the model probabilities, because of the quasilikelihood. But the multinomial likelihood is preferred here.

To do a multinomial model is quite easy. It's just a special case of a log-linear model:

test the statistical significance of the x group indicator term.

• 1 $\begingroup$ This is a good solution, especially considering that the experimental design is actually more complex than indicated in the original post. It would also allow comparisons among the individual treatments. I might add the following code: library(car); Anova(fit); library(emmeans); EM = emmeans(fit, ~ x, type="response"); EM; pairs(EM) $\endgroup$ –  Sal Mangiafico Commented Oct 22, 2018 at 20:14

Need to install package EMT.

It is called as goodness of fit test for multinomial distribution.

• $\begingroup$ Thank you so much for your answer. It seams that this is exactly the test I was looking for. But now I still have a little problem, because I get an error message, when I thry to run the model. This is my input, like you suggested: observed<- c(data1$SummeF, data1$SummeC, data1$SummeD) prob<- c(0.33333, 0.33333, 0.33333) out<- multinomial.test(observed, prob) And this the error message: Error in multinomial.test(observed, prob) : Observations and probabilities must have same dimensions. Thank you for your patience! $\endgroup$ –  R. Kienzle Commented Oct 22, 2018 at 13:07
• $\begingroup$ Observed has to counts, as in the example. $\endgroup$ –  Sal Mangiafico Commented Oct 22, 2018 at 14:09
• $\begingroup$ Maybe you forgot two dollar symbols observed<- c(data1SummeF,data1SummeC, data1$SummeD). Between data1 and SummerF, add a dollar symbol. I cannot type dollar symbol. $\endgroup$ –  user158565 Commented Oct 22, 2018 at 14:20
• $\begingroup$ No, the dollar symbols are in. The just did not survived the copy-paste, cause I can not type them either. $\endgroup$ –  R. Kienzle Commented Oct 22, 2018 at 14:59
• $\begingroup$ just type in c(data1SummeF,data1SummeC, data1$SummeD) to see if you can get three numbers. (With dollars.) $\endgroup$ –  user158565 Commented Oct 22, 2018 at 15:49

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• What happened to the genealogical records of the tribe of Judah and the house of David after Jerusalem was destroyed by the Romans in 70 A.D
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• Is it illegal to allow (verbally) someone to do action that may kill them?
• If Venus had a sapient civilisation similar to our own prior to global resurfacing, would we know it?