One-Sample Trimmed/Yuen Mean Test
Introduction
This test could be used to verify a single population mean (average). For example to test if the average age in a company is 34 years old. Since it only requires the scores on one variable, it is labelled a one-sample test.
As the name implies, this test uses a trimmed mean, rather than the regular mean. A trimmed mean simply removes some of the highest and lowest scores. It also uses a Winsorized variance, which replaces (not removes) some of the highest and lowest scores. The alternative Student t-test uses the regular mean.
Yuen developed the test for independent samples (to compare two means), but can also be used for one-sample scenario. The standard error can either be calculated using the first SE, which for example can be found in Tukey and McLaughlin (1963, p. 342), and seems similar to the independent samples version of this test as proposed by Yuen (1974, p. 167) The second version is used in the other libraries from the software R, and can be found in Wilcox (2012, p. 157), or Peró-Cebollero and Guàrdia-Olmos (2013, p. 409).
The advantage of trimming and using Windsorized variance, is that this test will be more suitable if the data comes from a non-normal distribution, which some claim to be a criteria to use the Student t-test.
Performing the Test
with Excel
Excel file from video: TS - Trimmed Mean (one-sample) (E).xlsm
with stikpetE
without stikpetE
with Python
Notebook from video: TS - Trimmed Mean (one-sample) (P).ipynb
with stikpetP
without stikpetP
Formulas
\(t = \frac{\bar{x}_t - \mu_{H_{0}}}{SE}\)
\(df = m - 1\)
\(sig = 2\times\left(1 - T\left(\left|t\right|, df\right)\right)\)
With:
\(\bar{x}_t = \frac{\sum_{i=g+1}^{n - g}y_i}{m}\)
\(g = \lfloor n\times p_{tes}\rfloor\)
\(m = n - 2\times g\)
\(SE_y = \sqrt{\frac{SSD_w}{m\times\left(m - 1\right)}}\)
\(SE_w = \frac{\sqrt{\frac{SSD_w}{n - 1}}}{\left(1 - 2\times p_t\right)\times\sqrt{n}}\)
\(SSD_w = g\times\left(y_{g+1} - \bar{x}_w\right)^2 + g\times\left(y_{n-g} - \bar{x}_w\right)^2 + \sum_{i=g+1}^{n - g} \left(y_i - \bar{x}_w\right)^2\)
\(\bar{x}_w = \frac{\bar{x}_t\times m + g\times\left(y_{g+1} + y_{n-g}\right)}{n}\)
The following symbols are used:
- \(\bar{x}_t\) the trimmed mean of the scores
- \(\bar{x}_w\) the Winsorized mean
- \(SSD_w\) the sum of squared deviations from the Winsorized mean
- \(n\) the sample size
- \(m\) the number of scores in the trimmed data set
- \(y\) the original scores but sorted from low to high
- \(p_{tes}\) the proportion of trimming on each side
- \(T\left(\dots\right)\) the cumulative distribution function of the Student t distribution
Two possible standard errors could be used. The first \(SE_y\) can be found in Tukey and McLaughlin (1963, p. 342) and is similar to what Yuen (1974, p. 167) uses for the independent samples. The second \(SE_w\) can be found in Wilcox (2012, p. 157), or Peró-Cebollero and Guàrdia-Olmos (2013, p. 409).
Interpreting the Result
The assumption about the population for this test (the null hypothesis) is that the mean is a specific value.
The test provides a p-value, which is the probability of a test statistic as from the sample, or even more extreme, if the assumption about the population would be true. If this p-value (significance) is below a pre-defined threshold (the significance level \(\alpha\) ), the assumption about the population is rejected. We then speak of a (statistically) significant result. The threshold is usually set at 0.05. Anything below is then considered low.
If the assumption is rejected, we conclude that the mean in the population will be different than the one used in the test.
Note that if we do not reject the assumption, it does not mean we accept it, we simply state that there is insufficient evidence to reject it.
Writing the results
Writing up the results of the test uses the format (APA, 2019 p. 182):
t(<degrees of freedom.>) = <t-value>, p = <p-value>
So for example:
The mean age of customers was 48.19 years (SD = 17.69). Using a 10% trimmed Yuen one-sample t-test, the claim that the average age is 50 years old can be rejected, t(1968) = -4.53, p < .001.
The p-value is shown with three decimal places, and no 0 before the decimal sign. If the p-value is below .0005, it can be reported as p < .001. The abbreviation of 'SD' for standard deviation, is not needed to be explained, it is an accepted abbreviation in APA (2019, p. 185). The name of the test is explicitly mentioned, since it is a somewhat obscure test to use.
APA (2019, p. 88) states to also report an effect size measure.
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