CALC - Regressions

R-squared Coefficient of Determination (R²)

R-squared Coefficient of Determination (R²): R-squared is a statistical measure that represents the proportion of the variance in the dependent variable that is explained by the independent variables in a regression model. It ranges from 0 to 1, with higher values indicating a better fit. Usually >0.95 is considered statistical correlation.

R2=1−∑i=1n(yi−y^i)2∑i=1n(yi−yˉ)2R^2 = 1 - \frac{\sum_{i=1}^{n} (y_i - \hat{y}i)^2}{\sum{i=1}^{n} (y_i - \bar{y})^2}

R2R^2 = Coefficient of determination yiy_i = Observed value y^i\hat{y}_i= Predicted value yˉ\bar{y} = Mean of observed values nn = Sample size

Spearman Rank Correlation Coefficient (p)

Spearman Rank Correlation Coefficient (p): Measures the strength and direction of a monotonic relationship between two variables. A minimum absolute value of 0.95 is standard for strong correlation.

p=1−6∑di2n(n2−1)p = 1 - \frac{6 \sum d_i^2}{n(n^2 - 1)}

did_i= Difference between ranks of X and Y nn = Number of pairs

Passing Bablok Correlation Coefficient (r)

Passing Bablok Correlation Coefficient (r): Measures the strength and direction of the linear relationship between two measurement methods using Passing Bablok regression. A minimum absolute value of 0.95 is standard for strong correlation.

r=Cov(X,Y)Var(X)+Var(Y)r = \frac{Cov(X, Y)}{Var(X) + Var(Y)}

Cov(X,Y)Cov(X, Y) = Covariance of X and Y

Var(X)Var(X) = Variance of X

Var(Y)Var(Y) = Variance of Y

Spearman Rank Correlation Coefficient (p)

Spearman Rank Correlation Coefficient (p): Measures the strength and direction of a monotonic relationship between two variables. A minimum absolute value of 0.95 is standard for strong correlation.

p=1−6∑di2n(n2−1)p = 1 - \frac{6 \sum d_i^2}{n(n^2 - 1)}

did_i= Difference between ranks of X and Y nn = Number of pairs

Deming Variance Ratio (vr)

Deming Variance Ratio (vr): Quantifies the agreement between two measurement methods, considering measurement errors. A maximum allowable difference from 1 is standard.

VR=Var(X)+Var(Y)Var(X−Y)VR = \frac{Var(X) + Var(Y)}{Var(X - Y)}

Var(X)Var(X) = Variance of X Var(Y)Var(Y) = Variance of Y

Sample Correlation Coefficient (R)

Sample Correlation Coefficient (R): The sample correlation coefficient (R) quantifies the strength and direction of the linear relationship between two variables based on sample data. A min value of 0.95 is standard.

R=r2R = \sqrt{r^2}

r2r^2 = Coefficient of Determination

Pearson Correlation Coefficient (r)

Pearson Correlation Coefficient (r): Measures the strength and direction of a linear relationship between two variables. A minimum absolute value of 0.95 is standard for strong correlation.

r=Cov(X,Y)σXσYr = \frac{Cov(X, Y)}{\sigma_X \sigma_Y}

Cov(X,Y)Cov(X, Y) = Covariance of X and Y σXσ_X = Standard Deviation of X σYσ_Y = Standard Deviation of Y

Cov(X,Y)=1n−1∑i=1n(xi−xˉ)(yi−yˉ)Cov(X, Y) = \frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x})(y_i - \bar{y})

nn = Number of data points xix_i = Individual data point of variable XyiXy_i = Individual data point of variable Y xˉ\bar{x} = Mean of variable X yˉ\bar{y} = Mean of variable Y