From: owner-ammf-digest@smoe.org (alt.music.moxy-fruvous digest) To: ammf-digest@smoe.org Subject: alt.music.moxy-fruvous digest V14 #5258 Reply-To: ammf@fruvous.com Sender: owner-ammf-digest@smoe.org Errors-To: owner-ammf-digest@smoe.org Precedence: bulk alt.music.moxy-fruvous digest Thursday, November 5 2020 Volume 14 : Number 5258 Today's Subjects: ----------------- our favorite Bluetooth speakers to gift this holiday season. ["Portable S] ---------------------------------------------------------------------- Date: Thu, 5 Nov 2020 11:45:51 -0500 From: "Portable Speaker" Subject: our favorite Bluetooth speakers to gift this holiday season. our favorite Bluetooth speakers to gift this holiday season. http://backwave.buzz/aiZOodgp5aJ_moYJtCDQFPtxYpC_0wzQoudA0thmmVVUaZjN http://backwave.buzz/A-rL_eiaWluO8ZCfB0y8cSt5Q0RUZaH5tnqjrBqG4ZsfBo2a The conventional dictum that "correlation does not imply causation" means that correlation cannot be used by itself to infer a causal relationship between the variables. This dictum should not be taken to mean that correlations cannot indicate the potential existence of causal relations. However, the causes underlying the correlation, if any, may be indirect and unknown, and high correlations also overlap with identity relations (tautologies), where no causal process exists. Consequently, a correlation between two variables is not a sufficient condition to establish a causal relationship (in either direction). A correlation between age and height in children is fairly causally transparent, but a correlation between mood and health in people is less so. Does improved mood lead to improved health, or does good health lead to good mood, or both? Or does some other factor underlie both? In other words, a correlation can be taken as evidence for a possible causal relationship, but cannot indicate what the causal relationship, if any, might be. Simple linear correlations Four sets of data with the same correlation of 0.816 The Pearson correlation coefficient indicates the strength of a linear relationship between two variables, but its value generally does not completely characterize their relationship. In particular, if the conditional mean of {\displaystyle Y}Y given {\displaystyle X}X, denoted {\displaystyle \operatorname {E} (Y\mid X)}{\displaystyle \operatorname {E} (Y\mid X)}, is not linear in {\displaystyle X}X, the correlation coefficient will not fully determine the form of {\displaystyle \operatorname {E} (Y\mid X)}{\displaystyle \operatorname {E} (Y\mid X)}. The adjacent image shows scatter plots of Anscombe's quartet, a set of four different pairs of variables created by Francis Anscombe. The four {\displaystyle y}y variables have the same mean (7.5), variance (4.12), correlation (0.816) and regression line (y = 3 + 0.5x). However, as can be seen on the plots, the distribution of the variables is very different. The first one (top left) seems to be distributed normally, and corresponds to what one would expect when considering two variables correlated and following the assumption of normality. The second one (top right) is not distributed normally; while an obvious relationship between the two variables can be observed, it is not linear. In this case the Pearson correlation coefficient does not indicate that there is an exact functional relationship: only the extent to which that relationship can be approximated by a linear relationship. In the third case (bottom left), the linear relationship is perfect, except for one outlier which exerts enough influence to lower the correlation coefficient from 1 to 0.816. Finally, the fourth example (bottom right) shows another example when one outlier is enough to produce a high correlation coefficient, even though the relationship between the two ------------------------------ End of alt.music.moxy-fruvous digest V14 #5258 **********************************************