Examples of least squares in the following topics:
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- The variability of points around the least squares line remains roughly constant.
- Should we have concerns about applying least squares regression to the Elmhurst data in Figure 7.12?
- Least squares regression can be applied to these data.
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- Calculate the least squares line.
- Calculate the least squares line.
- Calculate the least squares line.
- Calculate the least squares line.
- Calculate the least squares line.
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- The criteria for determining the least squares regression line is that the sum of the squared errors is made as small as possible.
- The criteria for the best fit line is that the sum of squared errors (SSE) is made as small as possible.
- Therefore, this best fit line is called the least squares regression line.
- Ordinary Least Squares (OLS) regression (or simply "regression") is a useful tool for examining the relationship between two or more interval/ratio variables assuming there is a linear relationship between said variables.
- This method minimizes the sum of squared vertical distances between the observed responses in the dataset and the responses predicted by the linear approximation.
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- Another popular estimation approach is the linear least squares method.
- The approach is called "linear" least squares since the assumed function is linear in the parameters to be estimated.
- In statistics, linear least squares problems correspond to a statistical model called linear regression which arises as a particular form of regression analysis.
- One basic form of such a model is an ordinary least squares model.
- Contrast why MLE and linear least squares are popular methods for estimating parameters
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- In this section, we use least squares regression as a more rigorous approach.
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- For the Elmhurst data, we could write the equation of the least squares regression line as
- The slope of the least squares line can be estimated by
- A common exercise to become more familiar with foundations of least squares regression is to use basic summary statistics and point-slope form to produce the least squares line.
- These lines should intersect on the least squares line.
- Summary of least squares fit for the Elmhurst data.
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- These points are especially important because they can have a strong influence on the least squares line.
- There are six plots shown in Figure 7.19 along with the least squares line and residual plots.
- For each scatterplot and residual plot pair, identify any obvious outliers and note how they influence the least squares line.
- In these cases, the outliers influenced the slope of the least squares lines.
- Six plots, each with a least squares line and residual plot.
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- A graph of averages and the least-square regression line are both good ways to summarize the data in a scatterplot.
- The most common method of doing this is called the "least-squares" method.
- The least-squares regression line is of the form $\hat{y} = a+bx$, with slope $b = \frac{rs_y}{s_x}$ ($r$ is the correlation coefficient, $s_y$ and $s_x$ are the standard deviations of $y$ and $x$).
- The points on a graph of averages do not usually line up in a straight line, making it different from the least-squares regression line.
- The graph of averages plots a typical $y$ value in each interval: some of the points fall above the least-squares regression line, and some of the points fall below that line.
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- However, it is more common to explain the strength of a linear fit using R2, called R-squared.
- The R2 of a linear model describes the amount of variation in the response that is explained by the least squares line.
- However, if we apply our least squares line, then this model reduces our uncertainty in predicting aid using a student's family income.
- This corresponds exactly to the R-squared value:
- Gift aid and family income for a random sample of 50 freshman students from Elmhurst College, shown with the least squares regression line.
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- This is where the chi-square distribution becomes useful.
- Each expected count must be at least 5.
- If each expected count is at least 5 and the null hypothesis is true, then the test statistic below follows a chi-square distribution with k − 1 degrees of freedom:
- Sample size / distribution: Each particular scenario (i.e. cell count) must have at least 5 expected cases.
- Degrees of freedom: We only apply the chi-square technique when the table is associated with a chi-square distribution with 2 or more degrees of freedom.