Analyse a Single Scale Variable

Excel file: IM - Binning.xlsm.

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without stikpetE:

Jupyter Notebook used in video: IM - Binning (nr of bins).ipynb.

with stikpetP:

Notebook from video: IM - Binning (nr of bins) (R).ipynb

with stikpetR:

without stikpetR:

For the formula's below \(k\) is the number of bins, and \(n\) the sample size. The \(\left \lceil \dots \right \rceil\) is the ceiling function, which means to round up to the nearest integer

Square-Root Choice (Duda & Hart, 1973)

\( k = \left \lceil\sqrt{n}\right \rceil \)

Sturges (1926, p. 65)

\( k = \left\lceil\log_2\left(n\right)\right\rceil+1 \)

QuadRoot (anonymous as cited in Lohaka, 2007, p. 87)

\( k = 2.5\times\sqrt[4]{n} \)

Rice (Lane, n.d.)

\( k = \left\lceil 2\times\sqrt[3]{n} \right\rceil \)

Terrell and Scott (1985, p. 209)

\( k = \sqrt[3]{2\times n} \)

Exponential(Iman & Conover as cited in Lohaka, 2007, p. 87)

\( k = \left\lceil\log_2\left(n\right)\right\rceil \)

Velleman (Velleman, 1976 as cited in Lohaka 2007)

\( \begin{cases}2\times\sqrt{n} & \text{ if } n\leq 100 \\ 10\times\log_{10}\left(n\right) & \text{ if } n > 100\end{cases} \)

Doane(1976, p. 182)

\( k = 1 + \log_2\left(n\right) + \log_2\left(1+\frac{\left|g_1\right|}{\sigma_{g_1}}\right) \)

In the formula's \(g_1\) the 3rd moment skewness:

\( g_1 = \frac{\sum_{i=1}^n\left(x_i-\bar{x}\right)^3} {n\times\sigma^3} = \frac{1}{n}\times\sum_{i=1}^n\left(\frac{x_i-\bar{x}}{\sigma}\right)^3 \)

With \(\sigma = \sqrt{\frac{\sum_{i=1}^n\left(x_i-\bar{x}\right)^2}{n}}\)

The \(\sigma_{g_1}\) is defined using the formula:

\( \sigma_{g_1}=\sqrt{\frac{6\times\left(n-2\right)}{\left(n+1\right)\left(n+3\right)}} \)

Formula's that determine the width (h) for the bins

Using the width and the range, it can be used to determine the number of categories:

\( k = \frac{\text{max}\left(x\right)-\text{min}\left(x\right)}{h} \)

Scott (1979, p. 608)

\( h = \frac{3.49\times s}{\sqrt[3]{n}} \)

Where \(s\) is the sample standard deviation:

\(s = \sqrt{\frac{\sum_{i=1}^n\left(x_i-\bar{x}\right)^2}{n-1}}\)

Freedman-Diaconis (1981, p. 3)

\( h = 2\times\frac{\text{IQR}\left(x\right)}{\sqrt[3]{n}} \)

Where \( \text{IQR}\) the inter-quartile range.

A more complex technique is for example from Shimazaki and Shinomoto (2007). For this, we define a 'cost function' that needs to be minimized:

\( C_k = \frac{2\times\bar{f_k}-\sigma_{f_k}}{h^2} \)

With \(\bar{f_k}\) being the average of the frequencies when using \(k\) bins, and \(\sigma_{f_k}\) the population variance. In formula notation:

\(\bar{f_k}=\frac{\sum_{i=1}^k f_{i,k}}{k}\)

\(\sigma_{f_k}=\frac{\sum_{i=1}^k\left(f_{i,k}-\bar{f_k}\right)^2}{k}\)

Where \(f_{i,k}\) is the frequency of the i-th bin when using k bins.

Note that if the data are integers, it is recommended to use also bin widths that are integers.

Stone (1984, p. 3) is similar, but uses as a cost function:

\(C_k = \frac{1}{h}\times\left(\frac{2}{n-1}-\frac{n+1}{n-1}\times\sum_{i=1}^k\left(\frac{f_i}{n}\right)^2\right)\)

Knuth (2019, p. 8) suggest to use the k-value that maximizes:

\(P_k=n\times\ln\left(k\right) + \ln\Gamma\left(\frac{k}{2}\right) - k\times\ln\Gamma\left(\frac{1}{2}\right) - \ln\Gamma\left(n+\frac{k}{2}\right) + \sum_{i=1}^k\ln\Gamma\left(f_i+\frac{1}{2}\right)\)

Excel file from videos available here.

with stikpetE add-in

without add-ins

Jupyter Notebook of video is available here.

with stikpetP library

without stikpetP library

with stikpetP library

Jupyter Notebook of video is available here.

without stikpetR library

R script of video is available here.

Datafile used in video: GSS2012-Adjusted.sav

There are a three different ways to create a frequency table with SPSS.

An SPSS workbook with instructions of the first two can be found here.

Excel file from video: VI - Histogram (single) (E).xlsx

with equal bin widths (2016 or later)

with equal bin widths (before v. 2016)

with unequal bin widths

Notebook from video: VI - Histogram (P).ipynb

using stikpetP

without using stikpetP

Notebook from video: VI - Histogram (R).ipynb

using stikpetR

without using stikpetR

Four different methods to get a histogram with SPSS, using Chart-builder, Legacy Dialogs, Frequencies, or Explore. A video for each is below.

using Chart-builder

using Legacy Dialogs

using Frequencies

using Explore

Excel file from video: CEDI - Mean and Standard Deviation.xlsm.

R script from video: CEDI - Mean and Standard Deviation.R.

There are a four different ways to determine the mean and standard deviation with SPSS.

The formula for the (arithmetic) mean is:

\(\bar{x} = \frac{\sum_{i=1}^n x_i}{n}\)

The formula for the (ubiased sample) standard deviation is:

\(s=\sqrt{\frac{\sum_{i=1}^n \left(x_i - \bar{x}\right)^2}{n-1}}\)

Where \(x_i\) is the i-th score, and \(n\) the sample size.

If your data is the entire population, the standard deviation is calculated by dividing by n instead of n - 1.

Excel file from video: TS - Student t (one-sample) (E).xlsm

with stikpetE

without stikpetE

A flowgorithm for the one-sample Student t-test in Figure 1.

It takes as input the scores, the hypothesized mean, and a string for which output to show (df, statistic, or p-value).

It uses a function for the mean (CEmean), standard deviation (VAsd) and the t cumulative distribution (DItcdf). These in turn require the MAsumReal function (sums an array of real values) and the standard normal cumulative distribution (DIsncdf).

Flowgorithm file: FL-TSstudentOS.fprg.

Notebook from video: TS - Student t (one-sample) (P).ipynb

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without stikpetP

Notebook from video: TS - Student t (one-sample) (R).ipynb

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Formula's

The t-value can be determined using:

\(t=\frac{\bar{x}-\mu_{H_{0}}}{SE}\)

In this formula \(\bar{x}\) is the sample mean, \(\mu_{H_0}\) the expected mean in the population (the mean according to the null hypothesis), and SE the standard error.

The standard error can be calculated using:

\(SE=\frac{s}{\sqrt{n}}\)

Where n is the sample size, and s the sample standard deviation.

The formula for the sample standard deviation is:

\(s=\sqrt{\frac{\sum_{i=1}^{n}\left(x_{i}-\bar{x}\right)^{2}}{n-1}}\)

In this formula x_i is the i-th score, and n is the sample size.

The sample mean can be calculated using:

\(\bar{x}=\frac{\sum_{i=1}^{n}x_{i}}{n}\)

The degrees of freedom is determined by:

\(df=n-1\)

Where n is the sample size

Example (different example)

We are given the ages of five students, and have an hypothesized population mean of 24. The ages of the students are:

\(X=\left\{18,21,22,19,25\right\}\)

Since there are five students, we can also set n = 5, and the hypothesized population of 24 gives \(\mu_{H_{0}}=24\)

For the standard deviation, we first need to determine the sample mean:

\(\bar{x}=\frac{\sum_{i=1}^{n}x_{i}}{n}=\frac{\sum_{i=1}^{5}x_{i}}{5}=\frac{18+21+22+19+25}{5}\)

\(=\frac{105}{5}=21\)

Then we can determine the standard deviation:

\(s=\sqrt{\frac{\sum_{i=1}^{n}\left(x_{i}-\bar{x}\right)^{2}}{n-1}}=\sqrt{\frac{\sum_{i=1}^{5}\left(x_{i}-21\right)^{2}}{5-1}}=\sqrt{\frac{\sum_{i=1}^{5}\left(x_{i}-21\right)^{2}}{4}}\)

\(=\sqrt{\frac{\left(18-21\right)^{2}}{4}+\frac{\left(21-21\right)^{2}}{4}+\frac{\left(22-21\right)^{2}}{4}+\frac{\left(19-21\right)^{2}}{4}+\frac{\left(25-21\right)^{2}}{4}}\)

\(=\sqrt{\frac{\left(-3\right)^{2}}{4}+\frac{\left(0\right)^{2}}{4}+\frac{\left(1\right)^{2}}{4}+\frac{\left(-2\right)^{2}}{4}+\frac{\left(4\right)^{2}}{4}}\)

\(=\sqrt{\frac{9}{4}+\frac{0}{4}+\frac{1}{4}+\frac{4}{4}+\frac{16}{4}}=\sqrt{\frac{9+0+1+4+16}{4}}=\sqrt{\frac{30}{4}}\)

\(=\sqrt{\frac{15}{2}}=\frac{1}{2}\sqrt{15\times2}=\frac{1}{2}\sqrt{30}\approx2.74\)

The standard error then becomes:

\(SE=\frac{s}{\sqrt{n}}=\frac{\frac{1}{2}\sqrt{30}}{\sqrt{5}}=\frac{\frac{\sqrt{30}}{2}}{\sqrt{5}}=\frac{\sqrt{30}}{2\times\sqrt{5}}\)

\(=\frac{1}{2}\times\frac{\sqrt{30}}{\sqrt{5}}=\frac{1}{2}\times\sqrt\frac{30}{5}=\frac{1}{2}\sqrt{6}\approx1.22\)

The t-value:

\(t=\frac{\bar{x}-\mu_{H_{0}}}{SE}=\frac{21-24}{\frac{1}{2}\sqrt{6}}=\frac{-3}{\frac{\sqrt{6}}{2}}=\frac{-3\times2}{\sqrt{6}}=\frac{-6}{\sqrt{6}}\)

\(gif.latex?=\frac{-6}{\sqrt{6}}\times\frac{\sqrt{6}}{\sqrt{6}}=\frac{-6\times\sqrt{6}}{\sqrt{6}\times\sqrt{6}}=\frac{-6\times\sqrt{6}}{6}=-\sqrt{6}\approx-2.45\)

The degrees of freedom is relatively simple:

\(df=n-1=5-1=4\)

The two-sided significance is then usually found by using the t-value and the df, and consulting a t-distribution table, or using some software. If you really had to determine it manually, it would involve the formula for the t-distribution (the cumulative density function):

\(2\times\int_{x=|t|}^{\infty}\frac{\Gamma\left(\frac{df+1}{2}\right)}{\sqrt{df\times\pi}\times\Gamma\left(\frac{df}{2}\right)}\times\left(1+\frac{x^2}{df}\right)^{-\frac{df+1}{2}}\)

See the Student t-distribution section for more details.

Excel file: ES - Hedges g (one-sample).xlsm.

without stikpetE add-in

A flowgorithm for Hedges g (one-sample) in Figure 1.

It takes as input paramaters the scores, the hypothesized mean, and which approximation to use (if any).

It uses a function for Cohen's d (EScohenDos), the gamma function (MAgamma), and the e power function (MAexp). Cohen's d function requires in turn the mean (CEmean) and standard deviation (VAsd) functions, which make use of the MAsumReal (sum of real values) function. The gamma function needs the factorial function (MAfact)

Flowgorithm file: FL-EShedgesGos.fprg.

Jupyter Notebook used in video: ES - Hedges g (one-sample) (P).ipynb.

without stikpetP library

Jupyter Notebook used in video: ES - Hedges g (one-sample) (R).ipynb.

video to be uploaded

This has become available in SPSS 27. There is no option in earlier versions of SPSS to determine Hedges g, but it can easily be calculated using the 'Compute variable' option as shown in the video.

The exact formula for Hedges g is (Hedges, 1981, p. 111):

\(g = d\times\frac{\Gamma\left(m\right)}{\Gamma\left(m-0.5\right)\times\sqrt{m}}\)

With \(d\) as Cohen's d, and \(m = \frac{df}{2}\), where \(df=n-1\)

The \(\Gamma\) indicates the gamma function, defined as:

\(\Gamma\left(a=\frac{x}{2}\right)=\begin{cases} \left(a - 1\right)! & \text{ if } x \text{ is even}\\ \frac{\left(2\times a\right)!}{4^a\times a!}\times\sqrt{\pi} & \text{ if } x \text{ is odd}\end{cases}\)

Because the gamma function is computational heavy there are a few different approximations.

Hedges himself proposed (Hedges, 1981, p. 114):

\(g^* \approx d\times\left(1 - \frac{3}{4\times df-1}\right)\)

Durlak (2009, p. 927) shows another factor to adjust Cohens d with:

\(g^* = d\times\frac{n-3}{n-2.25}\times\sqrt{\frac{n-2}{n}}\)

Xue (2020, p. 3) gave an improved approximation using:

\(g^* = d\times\sqrt[12]{1-\frac{9}{df}+\frac{69}{2\times df^2}-\frac{72}{df^3}+\frac{687}{8\times df^4}-\frac{441}{8\times df^5}+\frac{247}{16\times df^6}}\)

Note 1. You might come across the following approximation: (Hedges & Olkin, 1985, p. 81)

\(g^* = d\times\left(1-\frac{3}{4\times n - 9}\right)\)

This is an alternative but equal method, but for the paired version. In the paired version we set df = n - 2, and if we substitute this in the Hedges version it gives the same result. The denominator of the fraction will then be:

\(4\times df - 1 = 4 \times\left(n-2\right) - 1 = 4\times n-4\times2 - 1 = 4\times n - 8 - 1 = 4\times n - 9\)

Note 2. Often for the exact formula 'm' is used for the degrees of freedom, and the formula will look like:

\(g = d\times\frac{\Gamma\left(\frac{m}{2}\right)}{\Gamma\left(\frac{m-1}{2}\right)\times\sqrt{\frac{m}{2}}}\)

Note 3: Alternatively the natural logarithm gamma can be used by using:

\(j = \ln\Gamma\left(m\right) - 0.5\times\ln\left(m\right) - \ln\Gamma\left(m-0.5\right)\)

and then calculate:

\(g = d\times e^{j}\)

In the formula for \(j\) we still use \(m = \frac{df}{2}\).