Inferential statistics are concerned with making inferences based on relations found in the sample, to relations in the population. Inferential statistics help us decide, for example, whether the differences between groups that we see in our data are strong enough to provide support for our hypothesis that group differences exist in general, in the entire population. We will start by considering the basic principles of significance testing: the sampling and test statistic distribution, p-value, significance level, power and type I and type II errors. Then we will consider a large number of statistical tests and techniques that help us make inferences for different types of data and different types of research designs. For each individual statistical test we will consider how it works, for what data and design it is appropriate and how results should be interpreted. You will also learn how to perform these tests using freely available software. For those who are already familiar with statistical testing: We will look at z-tests for 1 and 2 proportions, McNemar's test for dependent proportions, t-tests for 1 mean (paired differences) and 2 means, the Chi-square test for independence, Fisher’s exact test, simple regression (linear and exponential) and multiple regression (linear and logistic), one way and factorial analysis of variance, and non-parametric tests (Wilcoxon, Kruskal-Wallis, sign test, signed-rank test, runs test).
4.01 Regression model
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Multiple regression
In this module we’ll see how we can use more than one predictor to describe or predict a quantitative outcome variable. In the social sciences relations between psychological and social variables are generally not very strong, since outcomes are generally influences by complex processes involving many variables. So it really helps to be able to describe an outcome variable with several predictors, not just to increase the fit of the model, but also to assess the individual contribution of each predictor, while controlling for the others.
与讲师见面
Annemarie Zand ScholtenAssistant Professor
Economics and Business
Emiel van LoonAssistant Professor
Institute for Biodiversity and Ecosystem Dynamics
In multiple regression, we use not one but
several quantitative predictors to predict a quantitative response variable.
In this video, you'll learn why multiple regression is useful and
how we express a multiple regression model at the sample and
population level, and how to interpret the regression coefficients and intercept.
Earlier, I tried to predict the popularity of cat videos
measured by number of page views using cat age.
If I were to collect actual data,
I would probably find that cat age doesn't predict popularity very well.
This is no wonder, the popularity of cat videos is
undoubtedly influenced by a lot of other things besides the cat's age.
Possible other relative characteristics are the cat's fluffiness or
hairiness in terms of hair length, its attractiveness,
how funny its behavior is, or to what extent it mimics human emotions.
With multiple regression, I can add such variables as additional predictors,
which will hopefully result in a more appropriate better-fitting model and
better predictions.
I can also add variables to control for their possibly confounding influence.
For example,
the time a video has been available online will influence its popularity.
It has nothing to do with the attractiveness of the video, but
it might explain why some videos of very cute and funny kittens aren't as popular
as expected, and some videos of older cats are more popular than expected.
Okay, so what does the model look like?
Well, it's an extension of the simple linear model at the sample level,
we express it as y hat sub i equals a + b sub
1 times x sub i sub 1 + b sub 2 times
x sub i sub 2 and so on, until we reach the last predictor, the noted m.
We end with b sub m times x sub i sub m.
Note that the sub i's indicated y, and the x's stand for individual values.
At the population level, we express the model as mu sub y = alpha +
beta sub 1 * x sub 1 + beta sub 2 * x sub 2 and
and so on, ending with beta sub m * x sub m.
To understand how to interpret this model,
let's consider a simple example with only two predictors.
Suppose we add hairiness as a predictor to model video popularity.
Hairiness is rated on a scale between 0 and 10,
with 0 meaning hairless and a 10 meaning long-haired like a Persian cat.
Say we find this regression equation, y hat sub i equals
34.372 minus 1.775 times age
sub i plus 1.414 times hairiness sub i
We can visualize this model by considering the relation between cat age and
video popularity at particular values of hairiness.
Say we take hairless cats with a hairiness score of zero.
Given this hairiness score, what is the relation between age and popularity?
Well, if we fill in zero in the equation,
we simple get Y hat sub i = 34.372- 1.775 times age sub i.
This can be drawn as a simple regression line.
Now consider the relation for a hairiness score of one.
If we enter a hairiness score of one in the equation, we get y hat sub i = 34.372.
- 1.775 times age sub i + 1.414,
which equals 35.786- 1.775 times age sub i.
If we enter a hairiness score of 2, we get y hat sub i equals 34.372-
1.775 times age sub i Plus 2.828 which equals 37.200 minus 1.775 times
age sub i. The regression lines predicting popularity with cat age,
at given values of hairiness all over on parallel. From this,
we can see that a multiple regression for a particular predictor,
the regression coefficient gives you the change in the response variable
per unit increase of that predictor given the values of the other predictors.
It's important to note that the size of each regression coefficient depends on
the scale of the predictor.
So we can't say that the predictor age,
which is larger, is more influential in predicting popularity than the hairiness.
Age ranges from 0 to about 50 while hairiness ranges between 0 and 10.
Another thing to note is that the value of the regression coefficient for
age in our multiple regression equation is different from the value
in the simple regression equation, even though the observations are the same.
In the simple case, with just a.
Age as predictor, we consider the relation between cat age and
popularity while ignoring all other variables.
By adding hairiness as a predictor, we control for the effective hairiness
when we consider the relation between cat age and popularity.
We consider the relation for each level of hairiness
which might result in a stronger or weaker relation between cat age and popularity.
We can visualize the entire model by adding another axis,
the z axis to represent hairiness.
You can see that the parallel lines now form a plane in a three dimensional graph.
This plane represents the predicted values produced by the model.
The intercept A is where the plane crosses the y-axis, so the intercept
A represents the predicted value when cat age and hairiness are both 0.
Just like in simple linear regression, I can calculate the residuals,
the vertical distances between the observations and
the predicted values, which in this case lie on the regression plane.
These residuals are used to find the intercept and regression coefficient
That provide the best fitting plane through the data points.
Just like in simple regression the residuals are minimized using the method
of ordinary least squares.
Because the results in formulas for the intercept in regression coefficients
are more complicated we'll use statistical software to calculate them.
At the population level we model the means of the conditional distributions,
mu sub y.
For every point on the plain, for every combination of cat age and
hairiness we assume there is a distribution of popularity scores.
The mean of this distribution lies on the plain.
The standard deviations of all these conditional distributions
are assumed to be identical, so the spread of observations around the plane
is assumed to be the same everywhere.
With more than two predictors,
it's no longer possible to represent the model visually in one graph.
But the logic and the interpretation of the intercept and
regression coefficients is the same.
