basic linear regression and multiple regression
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DESCRIPTIONBasic linear regression and multiple regression. Psych 437 - Fraley. Example. Let’s say we wish to model the relationship between coffee consumption and happiness. Some Possible Functions. Lines. Linear relationships Y = a + b X a = Y-intercept (the value of Y when X = 0) - PowerPoint PPT Presentation
Basic linear regression and multiple regressionPsych 437 - Fraley
ExampleLets say we wish to model the relationship between coffee consumption and happiness
Some Possible Functions
LinesLinear relationshipsY = a + bXa = Y-intercept (the value of Y when X = 0)b = slope (the rise over the run, the steepness of the line); a weightY = 1 + 2X
Lines and interceptsY = a + 2XNotice that the implied values of Y go up as we increase a.By changing a, we are changing the elevation of the line.
Y = 1 + 2XY = 3 + 2XY = 5 + 2X
Lines and slopesSlope as rise over run: how much of a change in Y is there given a 1 unit increase in X.As we move up 1 unit on X, we go up 2 units on Y2/1 = 2 (the slope)Y = 1 + 2Xrunrisemove from 0 to 1rise from 1 to 3 (a 2 unit change)
Lines and slopesNotice that as we increase the slope, b, we increase the steepness of the line COFFEEHAPPINESS-4-2024-50510Y = 1 + 2XY = 1 + 4X
Lines and slopesWe can also have negative slopes and slopes of zero.When the slope is zero, the predicted values of Y are equal to a. Y = a + 0X Y = a COFFEEHAPPINESS-4-2024-50510b=2b=4b=0b=-2b=-4
Other functionsQuadratic functionY = a + bX2a still represents the intercept (value of Y when X = 0)b still represents a weight, and influences the magnitude of the squaring function
Quadratic and interceptsAs we increase a, the elevation of the curve increases
COFFEEHAPPINESS-4-2024051015202530Y = 0 + 1X2Y = 5 + 1X2
Quadratic and WeightWhen we increase the weight, b, the quadratic effect is accentuatedCOFFEEHAPPINESS-4-2024020406080100120Y = 0 + 1X2Y = 0 + 5X2
Quadratic and WeightAs before, we can have negative weights for quadratic functions.In this case, negative values of b flip the curve upside-down.As before, when b = 0, the value of Y = a for all values of X.COFFEEHAPPINESS-4-2024-100-50050100Y = 0 5X2Y = 0 1X2Y = 0 + 1X2Y = 0 + 5X2Y = 0 + 0X2
Linear & Quadratic CombinationsWhen linear and quadratic terms are present in the same equation, one can derive j-shaped curves Y = a + b1X + b2X2
Some terminologyWhen the relation between variables are expressed in this manner, we call the relevant equation(s) mathematical modelsThe intercept and weight values are called parameters of the model. Although one can describe the relationship between two variables in the way we have done here, for now on well assume that our models are causal models, such that the variable on the left-hand side of the equation is assumed to be caused by the variable(s) on the right-hand side.
TerminologyThe values of Y in these models are often called predicted values, sometimes abbreviated as Y-hat or . Why? They are the values of Y that are implied by the specific parameters of the model.
EstimationUp to this point, we have assumed that our models are correct. There are two important issues we need to deal with, however:Assuming the basic model is correct (e.g., linear), what are the correct parameters for the model?Is the basic form of the model correct? That is, is a linear, as opposed to a quadratic, model the appropriate model for characterizing the relationship between variables?
EstimationThe process of obtaining the correct parameter values (assuming we are working with the right model) is called parameter estimation.
Parameter Estimation exampleLets assume that we believe there is a linear relationship between X and Y.Assume we have collected the following dataWhich set of parameter values will bring us closest to representing the data accurately?
Estimation exampleWe begin by picking some values, plugging them into the linear equation, and seeing how well the implied values correspond to the observed valuesWe can quantify what we mean by how well by examining the difference between the model-implied Y and the actual Y valuethis difference, , is often called error in prediction
Estimation exampleLets try a different value of b and see what happensNow the implied values of Y are getting closer to the actual values of Y, but were still off by quite a bit
Estimation exampleThings are getting better, but certainly things could improve
Estimation exampleAh, much better
Estimation exampleNow thats very niceThere is a perfect correspondence between the implied values of Y and the actual values of Y
Estimation exampleWhoa. Thats a little worse.Simply increasing b doesnt seem to make things increasingly better
Estimation exampleUgg. Things are getting worse again.
Parameter Estimation exampleHere is one way to think about what were doing: We are trying to find a set of parameter values that will give us a smallthe smallestdiscrepancy between the predicted Y values and the actual values of Y.How can we quantify this?
Parameter Estimation exampleOne way to do so is to find the difference between each value of Y and the corresponding predicted value (we called these differences errors before), square these differences, and average them together
Parameter Estimation exampleThe form of this equation should be familiar. Notice that it represents some kind of average of squared deviationsThis average is often called error variance.
Parameter Estimation exampleIn estimating the parameters of our model, we are trying to find a set of parameters that minimizes the error variance. In other words, we want to be as small as it possibly can be. The process of finding this minimum value is called least-squares estimation.
Parameter Estimation exampleIn this graph I have plotted the error variance as a function of the different parameter values we chose for b. Notice that our error was large at first (at b = -2), but got smaller as we made b larger. Eventually, the error reached a minimum when b = 2 and, then, began to increase again as we made b larger.Different values of b
Parameter Estimation exampleThe minimum in this example occurred when b = 2. This is the best value of b, when we define best as the value that minimizes the error variance. There is no other value of b that will make the error smaller. (0 is as low as you can go.)Different values of b
Ways to estimate parametersThe method we just used is sometimes called the brute force or gradient descent method to estimating parameters.More formally, gradient decent involves starting with viable parameter value, calculating the error using slightly different value, moving the best guess parameter value in the direction of the smallest error, then repeating this process until the error is as small as it can be.Analytic methodsWith simple linear models, the equation is so simple that brute force methods are unnecessary.
Analytic least-squares estimationSpecifically, one can use calculus to find the values of a and b that will minimize the error function
Analytic least-squares estimationWhen this is done (we wont actually do the calculus here ), the obtain the following equations:
Analytic least-squares estimationThus, we can easily find the least-squares estimates of a and b from simple knowledge of (1) the correlation between X and Y, (2) the SDs of X and Y, and (3) the means of X and Y:
A neat factNotice what happens when X and Y are in standard score form
In the parameter estimation example, we dealt with a situation in which a linear model of the form Y = 2 + 2X perfectly accounted for the data. (That is, there was no discrepancy between the values implied by the model and the actual data.)Even when this is not the case (i.e., when the model doesnt explain the data perfectly), we can still find least squares estimates of the parameters.
Error VarianceIn this example, the value of b that minimizes the error variance is also 2. However, even when b = 2, there are discrepancies between the predictions entailed by the model and the actual data values.
Thus, the error variance becomes not only a way to estimate parameters, but a way to evaluate the basic model itself.
R-squaredIn short, when the model is a good representation of the relationship between Y and X, the error variance of the model should be relatively low.
This is typically quantified by an index called the multiple R or the squared version of it, R2.
R-squared represents the proportion of the variance in Y that is accounted for by the modelWhen the model doesnt do any better than guessing the mean, R2 will equal zero. When the model is perfect (i.e., it accounts for the data perfectly), R2 will equal 1.00.
Neat factWhen dealing with a simple linear model with one X, R2 is equal to the correlation of X and Y, squared.
Why? Keep in mind that R2 is in a standardized metric in virtue of having divided the error variance by the variance of Y. Previously, when working with standardized scores in simple linear regression equations, we found that the parameter b is equal to r. Since b is estimated via least-squares techniques, it is directly related to R2.
Why is R2 useful? R2 is useful because it is a standard metric for interpreting model fit.It doesnt matter how large the variance of Y is because everything is evaluated relative to the variance of YSet end-points: 1 is perfect and 0 is as bad as a model can be.
Multiple RegressionIn many situations in personality psychology we are interested in modeling Y not only as a function of a single X variable, but potentially many X variables.
Example: We might attempt to explain variation in academic achievement as a function of SES and maternal education.
Y = a + b1*SES + b2*MATEDU
Notice that adding a new variable to the model is simple. This equation states that academic achievement is a function of at least two things, SES and MATEDU.
However, what the regression coefficients now represent is not merely the change in Y expected given a 1 unit increase in X. They represent the change in Y given a 1-unit change in X assuming all the other variables in the equation equal zero.In other words, these coefficients are kind of like partial correlations (technically, they are called semi-partial correlations). Were statistically controlling SES when estimating the effect of MATEDU.
Estimating regression coefficients in SPSS
Note: The regression parameter estimates are in the column labeled B. Constant = a = intercept
Achievement = 76.86 + 1.443*MATEDU + .539*SES
These parameter estimates imply that moving up one unit on SES leads to a 1.4 unit increase on achievement.
Moreover, moving up 1 unit in maternal education corresponds to a half-unit increase in achievement.
Does this mean that Maternal Education matters more than SES in predicting educational achievement?
Not necessarily. As it stands, the two variables might be on very different metrics. (Perhaps MATEDU ranges from 0 to 20 and SES ranges from 0 to 4.) To evaluate their relative contributions to Y, one can standardize both variables or examine standardized regression coefficients.
Z(Achievement) = 0 + .301*Z(MATEDU) + .118*Z(SES)
The multiple R and the R squared for the full model are listed here.
This particular model explains 14% of the variance in academic achievement
Adding SES*SES (SES2) improves R-squared by about 1%
These parameters suggest that higher SES predicts higher achievement, but in a limiting way. There are diminishing returns on the high end of SES.