### 统计代写|Generalized linear model代考广义线性模型代写|Linear Transformations and $z-S c o r e s$

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 统计代写|Generalized linear model代考广义线性模型代写|Linear Transformations

To perform a linear transformation, you need to take a dataset and either add, subtract, multiply, or divide every score by the same constant. (You may remember from Chapter 1 that a constant is a score that is the same for every member of a sample or population.) Performing a linear transformation is common in statistics. Linear transformations change the data, and that means they can have an impact on models derived from the data – including visual models and descriptive statistics.

Linear transformations allow us to convert data from one scale to a more preferable scale. For example, converting measurements recorded in inches into centimeters requires a linear transformation (multiplying by 2.54). Another common linear transformation is to convert Fahrenheit temperatures to Celsius temperatures (which requires subtracting 32 and then dividing by 1.8) or vice versa (which requires multiplying by $1.8$ and then adding 32 ). Sometimes performing a linear transformation is done to make calculations easier, as by eliminating negative numbers (by adding a constant so that all scores are positive) or eliminating decimals or fractions (by multiplying by a constant). As you will see in this chapter, making these changes in the scale of the data does not fundamentally alter the relative pattern of the scores or the results of many statistical analyses.

Linear transformations are common in statistics, so it is important to understand how linear transformations change statistics and data.

Impact of linear transformations on means. Chapter 4 showed how to calculate the mean (Formula 4.2) and standard deviation (Formulas 4.7-4.9). An interesting thing happens to these sample statistics when a constant is added to the data. If we use the Waite et al. (2015) data and add a constant of 4 to every person’s age, then the results will be as follows:
Finding the mean of these scores is as follows:
$$\begin{gathered} \bar{X}{\text {mew }}=\frac{22+23+24+25+27+27+28+28+29+35+37+39+42}{13} \ \bar{X}{\text {mew }}=\frac{386}{13}=29.69 \end{gathered}$$
The original mean (which we will abbreviate as $\bar{X}{\text {orig }}$ ) was 25.69, as shown in Guided Example 4.1. The difference between the original mean and the new mean is $4(29.69-25.69=4)$, which is the same value as the constant that was added to all of the original data. This is true for any constant that is added to every score in a dataset: $$\bar{X}{\text {orig }}+\text { constant }=\bar{X}_{\text {new }}$$
(Formula 5.1)

## 统计代写|Generalized linear model代考广义线性模型代写|A Special Linear Transformation

One common linear transformation in the social sciences is to convert the data to $z$-scores. A $z$-score is a type of data that has a mean of 0 and a standard deviation of 1 . Converting a set of scores to $z$ scores requires a formula:
$$z_{i}=\frac{x_{i}-\bar{X}}{s_{x}}$$
(Formula 5.7)
In Formula $5.7$ the $x_{i}$ refers to each individual score for variable $x, \bar{X}$ is the sample mean for variable $x$, and $s_{x}$ is the standard deviation for the same variable. The $i$ subscript in $z_{i}$ indicates that each individual score in a dataset has its own $z$-score.

There are two steps in calculating $z$-scores. The first step is in the numerator where the sample mean is subtracted from each score. (You may recognize this as the deviation score, which was discussed in Chapter 4 .) This step moves the mean of the data to 0 . The second step is to divide by the standard deviation of the data. Dividing by the standard deviation changes the scale of the data until the deviation is precisely equal to 1. Guided Practice $5.1$ shows how to calculate $z$-scores for real data.

The example $z$-score calculation in Guided Practice $5.1$ illustrates several principles of $z$-scores. First, notice how every individual whose original score was below the original mean (i.e., $\bar{X}=25.69$ ) has a negative $z$-score, and every individual whose original score was above the original mean has a positive $z$-score. Additionally, when comparing the original scores to the $z$-scores, it is apparent that the subjects whose scores are closest to the original mean also have $z$-scores closest to 0 which is the mean of the $z$-scores. Another principle is that the unit of $z$-scores is the standard deviation, meaning that the difference between each whole number is one standard deviation. Finally, in this example the person with the lowest score in the original data also has the lowest $z$-score – and the person with the highest score in the original data also has the highest $z$-score. In fact, the rank order of subjects’ scores is the same for both the original data and the set of $z$-scores because linear transformations (including converting raw scores to $z$-scores) do not change the shape of data or the ranking of individuals’scores. These principles also apply to every dataset that is converted to $z$-scores.
There are two benefits of $z$-scores. The first is that they permit comparisons of scores across different scales. Unlike researchers working in the physical sciences who have clear-cut units of

measurement (e.g., meters, grams, pounds), researchers and students in the social sciences frequently study variables that are difficult to measure (e.g., tension, quality of communication in a marriage). For example, a sociologist interested in gender roles may use a test called the Bem Sex Roles Inventory (Bem, 1974) to measure how strongly the subjects in a study identify with traditional masculine or feminine gender roles. This test has a masculinity subscale and a femininity subscale, each of which has scores ranging from 0 to 25 , with higher numbers indicating stronger identification with either masculine or feminine sex roles. A psychologist, though, may decide to use the Minnesota Multiphasic Personality Inventory’s masculinity and femininity subscales (Butcher, Dahlstrom, Graham, Tellegen, \& Kaemmer, 1989) to measure gender roles. Scores on these subscales typically range from 20 to 80 . Even though both researchers are measuring the same traits, their scores would not be comparable because the scales are so different. But if they convert their data to $z$-scores, then the scores from both studies are comparable.

Another benefit is that $z$-scores permit us to make comparisons across different variables. For example, an anthropologist can find that one of her subjects has a $z$-score of $+1.20$ in individualism and a $z$-score of $-0.43$ in level of cultural traditionalism. In this case she can say that the person is higher in their level of individualism than in their level of traditionalism. This example shows that comparing scores only requires that the scores be on the same scale. Because $z$-scores can be compared across scales and across variables, they function like a “universal language” for data of different scales and variables. For this reason $z$-scores are sometimes called standardized scores.

## 统计代写|Generalized linear model代考广义线性模型代写|Linear Transformations and Scales

In this chapter we have seen how linear transformations can be used to change the scale of the data. A common transformation is from the original data to $z$-scores, which have a mean of 0 and a standard deviation of 1 . We can change the scale into any form through applying linear transformations. Adding and subtracting a constant shifts data over to the desired mean, and multiplying and dividing by a constant condenses or expands the scale. This shows that all scales and axes in statistics are arbitrary (Warne, 2014a). This will be an important issue in later chapters. Regardless of how we change the scale, a linear transformation will never change the shape of the histogram of a dataset (and, consequently, its skewness and kurtosis).

We can change the scale of a distribution by performing a linear transformation, which is the process of adding, subtracting, multiplying, or dividing the data by a constant. Adding and subtracting a constant will change the mean of a variable, but not its standard deviation or variance. Multiplying and dividing by a constant will change the mean, the standard deviation, and the variance of a dataset. Table $5.1$ shows how linear transformations change the values of models of central tendency and variability.

One special linear transformation is the $z$-score, the formula for which is Formula 5.7. All $z$-score values have a mean of 0 and a standard deviation of 1 . Putting datasets on a common scale permits comparisons across different units. Linear transformations, like the $z$-score, force the data to have the mean and standard deviation that we want. Yet, they do not change the shape of the distribution – only its scale. In fact, all scales are arbitrary, and we can use linear transformations to give our data any mean and standard deviation we choose. We can also convert data from $z$-scores to any other scale using the linear transformation equation in Formula $5.8$.

## 统计代写|Generalized linear model代考广义线性模型代写|Linear Transformations

X¯喵喵 =22+23+24+25+27+27+28+28+29+35+37+39+4213 X¯喵喵 =38613=29.69

（公式 5.1）

(公式 5.7)

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## MATLAB代写

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