Basic Statistics and Data Analysis

Correlation Coeficient values lies between +1 and -1?

We know that the ratio of the explained variation to the total variation is called the coefficient of determination. This ratio is non-negative, therefore denoted by $r^2$, thus

\begin{align*}
r^2&=\frac{\text{Explained Variation}}{\text{Total Variation}}\\
&=\frac{\sum (\hat{Y}-\overline{Y})^2}{\sum (Y-\overline{Y})^2}
\end{align*}

It can be seen that if the total variation is all explained, the ratio $r^2$ (Coefficient of Determination) is one and if the total variation is all unexplained then the explained variation and the ratio r2 is zero.

The square root of the coefficient of determination is called the correlation coefficient, given by

\begin{align*}
r&=\sqrt{ \frac{\text{Explained Variation}}{\text{Total Variation}} }\\
&=\pm \sqrt{\frac{\sum (\hat{Y}-\overline{Y})^2}{\sum (Y-\overline{Y})^2}}
\end{align*}

and

$\sum (\hat{Y}-\overline{Y})^2=\sum(Y-\overline{Y})^2-\sum (Y-\hat{Y})^2$

therefore

\begin{align*}
r&=\sqrt{ \frac{\sum(Y-\overline{Y})^2-\sum (Y-\hat{Y})^2} {\sum(Y-\overline{Y})^2} }\\
&=\sqrt{1-\frac{\sum (Y-\hat{Y})^2}{\sum(Y-\overline{Y})^2}}\\
&=\sqrt{1-\frac{\text{Unexplained Variation}}{\text{Total Variation}}}=\sqrt{1-\frac{S_{y.x}^2}{s_y^2}}
\end{align*}

where $s_{y.x}^2=\frac{1}{n} \sum (Y-\hat{Y})^2$ and $s_y^2=\frac{1}{n} \sum (Y-\overline{Y})^2$

\begin{align*}
\Rightarrow r^2&=1-\frac{s_{y.x}^2}{s_y^2}\\
\Rightarrow s_{y.x}^2&=s_y^2(1-r^2)
\end{align*}

Since variances are non-negative

$\frac{s_{y.x}^2}{s_y^2}=1-r^2 \geq 0$

Solving for inequality we have

\begin{align*}
1-r^2 & \geq 0\\
\Rightarrow r^2 \leq 1\, \text{or}\, |r| &\leq 1\\
\Rightarrow & -1 \leq r\leq 1
\end{align*}

Alternative Proof

Since $\rho(X,Y)=\rho(X^*,Y^*)$ where $X^*=\frac{X-\mu_X}{\sigma_X}$ and $Y^*=\frac{Y-Y^*}{\sigma_Y}$

and as covariance is bi-linear and X* ,Y* have zero mean and variance 1, therefore

\begin{align*}
\rho(X^*,Y^*)&=Cov(X^*,Y^*)=Cov\{\frac{X-\mu_X}{\sigma_X},\frac{Y-\mu_Y}{\sigma_Y}\}\\
&=\frac{Cov(X-\mu_X,Y-\mu_Y)}{\sigma_X\sigma_Y}\\
&=\frac{Cov(X,Y)}{\sigma_X \sigma_Y}=\rho(X,Y)
\end{align*}

We also know that the variance of any random variable is ≥0, it could be zero i.e .(Var(X)=0) if and only if X is a constant (almost surely), therefore

$V(X^* \pm Y^*)=V(X^*)+V(Y^*)\pm2Cov(X^*,Y^*)$

As Var(X*)=1 and Var(Y*)=1, the above equation would be negative if $Cov(X^*,Y^*)$ is either greater than 1 or less than -1. Hence $1\geq \rho(X,Y)=\rho(X^*,Y^*)\geq -1$.

If $\rho(X,Y )=Cov(X^*,Y^*)=1$ then $Var(X^*- Y ^*)=0$ making X* =Y* almost surely. Similarly, if $\rho(X,Y )=Cov(X^*,Y^*)=-1$ then X*=−Y* almost surely. In either case, Y would be a linear function of X almost surely.

We can see that Correlation Coefficient values lies between -1 and +1.

Measure of Kurtosis

Kurtosis is a measure of peakedness of a distribution relative to the normal distribution. A distribution having a relatively high peak is called leptokurtic. A distribution which is flat topped is called platykurtic. The normal distribution which is neither very peaked nor very flat-topped is also called mesokurtic.  The histogram is an effective graphical technique for showing both the skewness and kurtosis of data set.

Data sets with high kurtosis tend to have a distinct peak near the mean, decline rather rapidly, and have heavy tails. Data sets with low kurtosis tend to have a flat top near the mean rather than a sharp peak.

Moment ratio and Percentile Coefficient of kurtosis are used to measure the kurtosis

Moment Coefficient of Kurtosis= $b_2 = \frac{m_4}{S^2} = \frac{m_4}{m^{2}_{2}}$

Percentile Coefficient of Kurtosis = $k=\frac{Q.D}{P_{90}-P_{10}}$
where Q.D = $\frac{1}{2}(Q_3 – Q_1)$ is the semi-interquartile range. For normal distribution this has the value 0.263.

A normal random variable has a kurtosis of 3 irrespective of its mean or standard deviation. If a random variable’s kurtosis is greater than 3, it is said to be Leptokurtic. If its kurtosis is less than 3, it is said to be Platykurtic.

Bias (Statistical Bias)

Bias is defined as the difference between the expected value of a statistic and the true value of the corresponding parameter. Therefore the bias is a measure of the systematic error of an estimator. The bias indicates the distance of the estimator from the true value of the parameter. For example, if we calculate the mean of large number of unbiased estimators, we will find the correct value.

Gauss, C.F. (1821) during his work on the least squares method gave the concept of an unbiased estimator.

Bias of an estimator of a parameter should not be confused with its degree of precision as degree of precision is a measure of the sampling error.

There are several types of bias which should not be considered as mutually exclusive

• Selection Bias (arise due to systematic differences between the groups compared)
• Exclusion Bias (arise due to the systematic exclusion of certain individuals from the study)
• Analytical Bias (arise due to the way that the results are evaluated)

Mathematically Bias can be Defined as

Let statistics T used to estimate a parameter θ, if E(T)=θ + b(θ) then b(θ) is called the bias of the statistic T, where E(T) represents the expected value of the statistics T. Note that if b(θ)=0, then E(T)=θ. So T is an unbiased estimator of θ.

Reference:
Gauss, C.F. (1821, 1823, 1826). Theoria Combinationis Observationum Erroribus Minimis Obnoxiae, Parts 1, 2 and suppl. Werke 4, 1-108.

Testing of Hypothesis

The researcher is similar to the prosecuting attorney is the sense that the researcher brings the null hypothesis “to trial” when she believes there is probability strong evidence against the null.

• Just as the prosecutor usually believes that the person on trial is not innocent, the researcher usually believes that the null hypothesis is not true.
• In the court system the jury must assume (by law) that the person is innocent until the evidence clearly calls this assumption into question; analogously, in hypothesis testing the researcher must assume (in order to use hypothesis testing) that the null hypothesis is true until the evidence calls this assumption into question.

Type I Error

It has become part of the statistical hypothesis testing culture.

• It is a longstanding convention.
• It reflects a concern over making type I errors (i.e., wanting to avoid the situation where you reject the null when it is true, that is, wanting to avoid “false positive” errors).
• If you set the significance level at .05, then you will only reject a true null hypothesis 5% or the time (i.e., you will only make a type I error 5% of the time) in the long run.