1. Foundations for Inference & Introduction to JASP

• Statistical inference is primarily concerned with understanding and quantifying the uncertainty of parameter estimates.
• Confidence Interval is a range of values where the true population value is likely to lie.

Foundations for Inference ^{1 3 4}

• Point Estimate: A single value that best approximates the population parameter.

Uncertainty in point estimates

• Sampling Error:
  • The natural variability that we expect between different random samples and the total population.
  • It results from the randomness of the sampling process.
  • It is quantified by the Standard Error as it is the most cared about measure of uncertainty.
• Bias:
  • The point estimate is systematically higher or lower than the population parameter.
  • It is a systemic tendency to under or over estimate the population parameter.
  • It is especially important during the data collection phase.

Sampling Distribution

• If we take many samples of the same size from the same population, the point estimates will vary from sample to sample.
• If we plot the distribution of point estimates, we get the sampling distribution.
• The distribution of point estimates based on samples of a fixed size from a certain population.
• It resembles a normal distribution (bell-shaped, symmetric) centered at the true population parameter.
• The mean of the sampling distribution is the population parameter.
• The standard deviation of the sampling distribution is the standard error.
• 

Central Limit Theorem

• When observations are independent and the sample size is sufficiently large, the sampling distribution of the parameter estimate will follow a normal distribution with a mean equal to the population parameter \({\mu}_{\hat{p}}=p\) and a stand error computed as \(SE_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}}\).
• For the CLT to hold, two conditions must be met
  • The independence of observations.
  • The success-failure condition:
    • \(np > 10\).
    • \(n(1-p) > 10\).
• The problem then becomes normal distribution problem:
  • The mean of the sampling distribution is the point estimate.
  • The standard error is the standard deviation of the sampling distribution.
  • We plot the distribution by finding the z-score and using the normal distribution table.
  • \(z_{1} = \frac{\hat{p_{min}} - p}{SE_{\hat{p}}}\), \(z_{2} = \frac{\hat{p_{max}} - p}{SE_{\hat{p}}}\).
  • 
  • It is hard to find z1 nad z2 as it requires us to make more samplings.
  • We use the confidence interval to estimate the range of values where the true population parameter is likely to lie.
  • With a 95% confidence level, we can say that \(z_{1} = -1.96\) and \(z_{2} = 1.96\).
  • With a 99% confidence level, we can say that \(z_{1} = -2.58\) and \(z_{2} = 2.58\).

The Plug-in Principle

• If we have a point estimate for a sample, and we confirmed that the central limit theorem holds and the sampling distribution is approximately normal, we can use the plug-in principle to plugin the sample point estimation (sample parameter) in place of the population parameter.

Confidence Intervals

• The problem is the point estimate (of a sample) may not truly represent the population parameter.
• Instead of providing a single point estimate, we provide a range of values where the true population parameter is likely to be.
• Constructing 95% confidence interval:
  • In a normal distribution, 95% of the observations fall within 1.96 standard deviations of the mean (distribution center).
  • Thus, if a point estimate can be modeled using a normal distribution, we can construct a plausible range with 95% confidence as \([\hat{p} - 1.96\times{SE}, \hat{p} + 1.96\times{SE}]\)
• Interpreting confidence level:
  • We are 95% confident that the true population parameter lies within the interval.
  • For example, [0.45, 0.55] is the 95% confidence level for people supporting solar panels, and we can say:
    • We are 95% confident that the actual percentage of public supporting solar panels is between 45% and 55%.
• Common confidence levels:
  • 90% confidence level: \(z = 1.645\) and the interval is \(\hat{p} \pm 1.645\times{SE}\).
  • 95% confidence level: \(z = 1.96\) and the interval is \(\hat{p} \pm 1.96\times{SE}\).
  • 99% confidence level: \(z = 2.58\) and the interval is \(\hat{p} \pm 2.58\times{SE}\).
• Confidence intervals says nothing about individual observations.
• Confidence intervals says nothing about future samples.
• It is NOT the probability that the true parameter lies within the interval.

Introduction to JASP ^{2 5}

• Data can be downloaded from https://osf.io/bx6uv/

Descriptive Statistics

• Descriptive statistics and related plots are a succinct way of describing and summarising data but do not test any hypotheses. it includes:
  • Measures of central tendency.
  • Measures of dispersion.
  • Percentile values.
  • Measures of distribution.
  • Descriptive plots.

Central Tendency

• The tendency for variable values to cluster around a central value.
• Mean, Median, Mode.
• Mean: M or \(\bar{x}\). It equals the sum of all values divided by the number of values. It equals the average. It is sensitive to outliers.
• Median: Mdn. It is the middle value when all values are ordered. It is less sensitive to outliers.
• Mode: The most frequent value.

Dispersion

• The spread of values around the central value.
• Standard Error of the mean, standard deviation, coefficient of variation, median absolute deviation, median absolute deviation Robust, inter-quartile range, variance, confident interval.
• Standard Error of the mean (SE):
  • It is the standard deviation of the sampling distribution of the mean.
  • It measures how far the sample mean from the true population mean.
  • It decreases as the sample size increases.
• Standard Deviation (SD):
  • It quantifies the amount of dispersion of the data around the mean.
  • Low SD means that values are close to the mean.
  • High SD means that values are far from the mean or dispersed on a wider range.
  • It is sensitive to outliers.
  • It is the square root of the variance.
  • It is the average distance of each data point from the mean.
  • It is the square root of the average of the squared differences between each data point and the mean.
• Coefficient of Variation (CV):
  • It measures the relative dispersion of the data.
  • It differs from the standard deviation which measures the absolute dispersion.
• Median Absolute Deviation (MAD):
  • It is the median of the absolute differences between each data point and the median.
  • It is less sensitive to outliers.
  • It only works with normally distributed data, otherwise, it is not recommended and SD is more useful.
• Median Absolute Deviation Robust (MAD Robust):
  • It is the median absolute deviation for the data but adjusted by a factor for asymptotically normal consistency.
• Inter-Quartile Range (IQR):
  • It is the difference between the 75^th percentile and the 25^th percentile.
  • It is similar to MAD but less robust.
• Variance:
  • It is the average of the squared differences between each data point and the mean.
  • It is the square of the standard deviation.
  • It is sensitive to outliers.
• Confidence Intervals:
  • It is a range of values where the true population value is likely to lie.

Quartiles

• Quartiles are where datasets are split into 4 equal quarters, normally based on rank ordering of median values.
• The 1^st quartile = 25^th percentile = lower quartile.
• The 2^nd quartile = 50^th percentile = median.
• The 3^rd quartile = 75^th percentile = upper quartile.
• The 4^th quartile = 100^th percentile = maximum value.
• The inter-quartile range = 3^rd quartile - 1^st quartile.
• The data is ordered like we do when computing the median, and then we pick the values at the 25^th, 50^th, and 75^th percentiles.

Distribution

• The distribution of data can be described by the shape, skewness, and kurtosis.
• In normal distribution, the skewness and kurtosis should be close to 0.
• Skewness:
  • It describes the shift of the distribution away from normal distribution.
  • Skewed distribution loses symmetry and the bell is shifted to the left or right.
  • Negative skewness show the mode moves to the right which causes left tail.
  • Positive skewness show the mode moves to the left which causes right tail.
  • 
• Kurtosis:
  • It describes how pointy or flat the bell is.
  • Positive kurtosis means the distribution is more peaked than a normal distribution., and tails are smaller.
  • Negative kurtosis means the distribution is less peaked than a normal distribution, and tails are larger.
  • 
• Shapiro-Wilk Test:
  • It is a test for normality, it assesses weather the data is significantly different from a normal distribution.

References

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1. Diez, D. M., Barr, C. D., & Çetinkaya-Rundel, M. (2019). Openintro statistics - Fourth edition. Open Textbook Library. https://www.biostat.jhsph.edu/~iruczins/teaching/books/2019.openintro.statistics.pdf Read Chapter 5 - Foundations for Inference from page 168-205. Section 5.1 - Point estimates and sampling variability. Section 5.2 - Confidence intervals for a proportion. Solve the following practice exercises as homework from the attached: Practice Exercises – Unit 1 https://my.uopeople.edu/pluginfile.php/1897551/mod_book/chapter/531355/Practice%20Excercises%20-%20%20Unit%201_Final.pdf ↩

2. Goss-Sampson, M. A. (2022). Statistical analysis in JASP: A guide for students (5^th ed., JASP v0.16.1 2022). https://jasp-stats.org/wp-content/uploads/2022/04/Statistical-Analysis-in-JASP-A-Students-Guide-v16.pdf Read Page 2-31 ↩

3. OpenIntroOrg. (2019a, September 02). Foundations for inference: Point estimates [Video]. YouTube. https://youtu.be/oLW_uzkPZGA ↩

4. OpenIntroOrg. (2019b, September 6). Intro to confidence intervals via proportions [Video]. YouTube. https://youtu.be/A6_W8qY8zJo ↩

5. JASP Statistics. (2022, October 05). Introduction to JASP [Video]. YouTube. https://youtu.be/APRaBFC2lEQ ↩