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

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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|Frequency Tables

One of the simplest ways to display data is in a frequency table. To create a frequency table for a variable, you need to list every value for the variable in the dataset and then list the frequency, which is the count of how many times each value occurs. Table 3.2a shows an example from Waite et al.’s (2015) study. When asked how much the subjects agree with the statement “I feel that my sibling understands me well,” one person strongly disagreed, three people disagreed, three felt neutral about the statement, five agreed, and one strongly agreed. A frequency table just compiles this information in an easy to use format, as shown below. The first column of a frequency table consists of the response label (labeled “Response”) and the second column is the number of each response (labeled “Frequency”).

Frequency tables can only show data for one variable at a time, but they are helpful for discerning general patterns within a variable. Table $3.2 \mathrm{a}$, for example, shows that few people in the Waite et al. (2015) study felt strongly about whether their sibling with autism understood them well. Most responses clustered towards the middle of the scale.

Table $3.2 \mathrm{a}$ also shows an important characteristic of frequency tables: the number of responses adds up to the number of participants in the study. This will be true if there are responses from everyone in the study (as is the case in Waite et al.’s study).

Frequency tables can include additional information about subjects’ scores in new columns. The first new column is labeled “Cumulative frequency,” which is determined by counting the number of people in a row and adding it to the number of people in higher rows in the frequency table. For example, in Table $3.2 \mathrm{~b}$, the cumulative frequency for the second row (labeled “Disagree”) is 4 – which was calculated by summing the frequency from that row, which was 3 , and the

frequency of the higher row, which was $1(3+1=4)$. As another example, the cumulative frequency column shows that, for example, 7 of the 13 respondents responded “strongly disagree,” “disagree,” or “neutral” when asked about whether their sibling with autism understood them well.

It is also possible to use a frequency table to identify the proportion of people giving each response. In a frequency table, a proportion is a number expressed as a decimal or a fraction that shows how many subjects in the dataset have the same value for a variable. To calculate the proportion of people who gave each response, use the following formula:
$$\text { Proportion }=\frac{f}{n} \quad \text { (Formula 3.1) }$$
In Formula $3.1, f$ is the frequency of a response, and $n$ is the sample size. Therefore, calculating the proportion merely requires finding the frequency of a response and dividing it by the number of people in the sample. Proportions will always be between the values of 0 to 1 .

Table $3.3$ shows the frequency table for the Waite et al. (2015) study, with new columns showing the proportions for the frequencies and the cumulative frequencies. The table nicely provides examples of the properties of proportions. For example, it is apparent that all of the values in both the “Frequency proportion” and “Cumulative proportion” columns are between 0 and 1.0. Additionally, those columns show how the frequency proportions were calculated – by taking the number in the frequency column and dividing by $n$ (which is 13 in this small study). A similar process was used to calculate the cumulative proportions in the far right column.

Table $3.3$ is helpful in visualizing data because it shows quickly which variable responses were most common and which responses were rare. It is obvious, for example, that “Agree” was the most common response to the question of whether the subjects’ sibling with autism understood them well, which is apparent because that response had the highest frequency (5) and the highest frequency proportion (.385). Likewise, “strongly disagree” and “strongly agree” were unusual responses because they tied for having the lowest frequency (1) and lowest frequency proportion (.077). Combined together, this information reveals that even though autism is characterized by pervasive difficulties in social situations and interpersonal relationships, many people with autism can still have a relationship with their sibling without a disability (Waite et al., 2015). This is important information for professionals who work with families that include a person with a diagnosis of autism.

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

Frequency tables are very helpful, but they have a major drawback: they are just a summary of the data, and they are not very visual. For people who prefer a picture to a table, there is the option of creating a histogram, an example of which is shown below in Figure 3.1.

A histogram converts the data in a frequency table into visual format. Each bar in a histogram (called an interval) represents a range of possible values for a variable. The height of the bar represents the frequency of responses within the range of the interval. In Figure $3.1$ it is clear that, for the second bar, which represents the “disagree” response, three people gave this response (because the bar is 3 units high).

Notice how the variable scores (i.e., 1, 2, 3, 4, or 5) are in the middle of the interval for each bar. Technically, this means that the interval includes more than just whole number values. This is illustrated in Figure 3.2, which shows that each interval has a width of 1 unit. The figure also annotates how, for the second bar, the interval ranges between $1.5$ and $2.5$. In Waite et al.’s study (2015), all of the responses were whole numbers, but this is not always true of social science data, especially for variables that can be measured very precisely, such as reaction time (often measured in milliseconds) or grade-point average (often calculated to two or three decimal places).

Figures $3.1$ and $3.2$ also show another important characteristic of histograms: the bars in the histogram touch. This is because the variable on this histogram is assumed to measure a continuous trait (i.e., level of agreement), even if Waite et al. (2015) only permitted their subjects to give five possible responses to the question. In other words, there are probably subtle differences in how much each subject agreed with the statement that their sibling with autism knows them well. If Waite et al. (2015) had a more precise way of measuring their subjects’ responses, it is possible that the data would include responses that were not whole numbers, such as $1.8,2.1$, and 2.3. However, if only whole numbers were permitted in their data, these responses would be rounded to 2. Yet, it is still possible that there are differences in the subjects’ actual levels of agreement. Having the histogram bars touch shows that, theoretically, the respondents’ opinions are on a continuum.

The only gaps in a histogram should occur in intervals where there are no values for a variable within the interval. An example of the types of gaps that are appropriate for a histogram is Figure $3.3$, which is a histogram of the subjects’ ages.

The gaps in Figure $3.3$ are acceptable because they represent intervals where there were no subjects. For example, there were no subjects in the study who were 32 years old. (This can be verified by checking Table 3.1, which shows the subjects’ ages in the third column.) Therefore, there is a gap at this place on the histogram in Figure 3.3.

Some people do not like having a large number of intervals in their histograms, so they will combine intervals as they build their histogram. For example, having intervals that span 3 years instead of 1 year will change Figure $3.3$ into Figure 3.4.

Combining intervals simplifies the visual model even more, but this simplification may be needed. In some studies with precisely measured variables or large sample sizes, having an interval for every single value of a variable may result in complex histograms that are difficult to understand. By simplifying the visual model, the data become more intelligible. There are no firm guidelines for choosing the number of intervals in a histogram, but I recommend having no fewer than five. Regardless of the number of intervals you choose for a histogram, you should always ensure that the intervals are all the same width.

## 统计代写|Generalized linear model代考广义线性模型代写|Number of Peaks

Another way to describe distributions is to describe the number of peaks that the histogram has. A distribution with a single peak is called a unimodal distribution. All of the distributions in the chapter so far (including the normal distribution in Figure $3.9$ and Quetelet’s data in Figure $3.10$ ) are unimodal distributions because they have one peak. The term “-modal” refers to the most common score in a distribution, a concept explained in further detail in Chapter $4 .$

Sometimes distributions have two peaks, as in Figure 3.17. When a distribution has two peaks, it is called a bimodal distribution. In the social sciences most distributions have only one peak, but bimodal distributions like the one in Figure $3.17$ occur occasionally, as in Lacritz et al.’s (2004) study that found that people’s test scores on a test of visual short-term memory were bimodal. In that case the peaks of the histograms were at the two extreme ends of the distribution – meaning people tended to have very poor or very good recognition of pictures they had seen earlier. Bimodal distributions can occur when a sample consists of two different groups (as when some members of a sample have had a treatment and the other sample members have not).

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

部分 =Fn （公式 3.1）

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## MATLAB代写

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