<div dir="ltr"><div dir="ltr"><div class="gmail_default" style="font-family:monospace,monospace">Dear Loet,<br></div><div class="gmail_default" style="font-family:monospace,monospace">I doubt that the square root of (observed - expected) makes sense: is the quantity inside parentheses always non-negative?</div><div class="gmail_default" style="font-family:monospace,monospace">May be you meant a chi-2 test on a 2x2 contingency table</div><div class="gmail_default" style="font-family:monospace,monospace"><a href="http://www.stat.yale.edu/Courses/1997-98/101/chisq.htm">http://www.stat.yale.edu/Courses/1997-98/101/chisq.htm</a></div><div class="gmail_default" style="font-family:monospace,monospace">Best regards,<br></div><div class="gmail_default" style="font-family:monospace,monospace">Michel.</div><div class="gmail_default" style="font-family:monospace,monospace"><br></div><div class="gmail_default" style="font-family:monospace,monospace">Michel Petitjean<br>Université de Paris, BFA, CNRS UMR 8251, INSERM ERL U1133, F-75013 Paris, France<br>Phone: +331 5727 8434; Fax: +331 5727 8372<br>E-mail: <a href="mailto:petitjean.chiral@gmail.com">petitjean.chiral@gmail.com</a> (preferred),<br> <a href="mailto:michel.petitjean@univ-paris-diderot.fr">michel.petitjean@univ-paris-diderot.fr</a><br><a href="http://petitjeanmichel.free.fr/itoweb.petitjean.html">http://petitjeanmichel.free.fr/itoweb.petitjean.html</a><br></div></div><br><div class="gmail_quote"><div dir="ltr" class="gmail_attr">Le dim. 5 juil. 2020 à 17:01, Loet Leydesdorff <<a href="mailto:loet@leydesdorff.net">loet@leydesdorff.net</a>> a écrit :<br></div><blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left:1px solid rgb(204,204,204);padding-left:1ex"><div><div id="gmail-m_1877032139338450650xca030115c06040e">Dear Marcus and colleagues: </div><div id="gmail-m_1877032139338450650xca030115c06040e"><br></div><div id="gmail-m_1877032139338450650xca030115c06040e"><br></div><div id="gmail-m_1877032139338450650xca030115c06040e"><blockquote cite="http://CACJqm9zq_JtFFeCM2NxDRP0N1eT3nZmdMP+h3Y9JW8xDEbVmwQ@mail.gmail.com" type="cite" class="gmail-m_1877032139338450650cite2"><div dir="ltr"><div dir="ltr"><div dir="ltr"><div><div dir="ltr"><div dir="ltr"><div dir="ltr">– I enjoyed your re-examination of Bateson’s `difference that makes a difference’. I seem to recall him also saying `differences themselves must be differentiated’. For my own purposes/thinking, I reframe this as `differentiated(/ing) differences’ which then ties to `levels’, and `levels of abstraction’ noted by Korzybski (although I have never see him articulating those levels?). Still, a `difference that makes a difference‘ I believe points to specific `meta levels’, but where `differentiated differences’ points to a meta-meta (general/universal/priamry) informatic level. This meta versus meta-meta perspective is yet another way of viewing (I believe) primary and secondary roles.</div><div dir="ltr"><br></div></div></div></div></div></div></div></blockquote>On second thought, it seems to me that a simple and most appropriate measure for the difference that one difference makes for another could be chi-square statistics.</div><div id="gmail-m_1877032139338450650xca030115c06040e"><br></div><div id="gmail-m_1877032139338450650xca030115c06040e">For example, one can ask whether a specific treatment makes a difference differently for man and women. The two variables can be cross-tabled as in Table 1.</div><div id="gmail-m_1877032139338450650xca030115c06040e"><br><div id="gmail-m_1877032139338450650x6e34d26becf445d4ad84c1af0aceb2c0">
<table style="width:218pt" width="292" cellspacing="0" cellpadding="0" border="0">
<colgroup><col style="width:47pt" width="63">
<col style="width:61pt" width="82">
<col style="width:63pt" width="84">
<col style="width:47pt" width="63">
</colgroup><tbody><tr style="height:14pt" height="19">
<td class="gmail-m_1877032139338450650xl65" style="height:14pt;width:47pt" width="63" height="19">Observed</td>
<td class="gmail-m_1877032139338450650xl65" style="border-left:medium none;width:61pt" width="82">Treatment </td>
<td class="gmail-m_1877032139338450650xl65" style="border-left:medium none;width:63pt" width="84">No treatment</td>
<td style="width:47pt" width="63"></td>
</tr>
<tr style="height:14pt" height="19">
<td class="gmail-m_1877032139338450650xl65" style="height:14pt;border-top:medium none" height="19">male</td>
<td class="gmail-m_1877032139338450650xl65" style="border-top:medium none;border-left:medium none" align="right">156</td>
<td class="gmail-m_1877032139338450650xl65" style="border-top:medium none;border-left:medium none" align="right">63</td>
<td align="right">219</td>
</tr>
<tr style="height:14pt" height="19">
<td class="gmail-m_1877032139338450650xl65" style="height:14pt;border-top:medium none" height="19">female</td>
<td class="gmail-m_1877032139338450650xl65" style="border-top:medium none;border-left:medium none" align="right">86</td>
<td class="gmail-m_1877032139338450650xl65" style="border-top:medium none;border-left:medium none" align="right">74</td>
<td align="right">160</td>
</tr>
<tr style="height:14pt" height="19">
<td style="height:14pt" height="19"></td>
<td align="right">242</td>
<td align="right">137</td>
<td align="right">379</td>
</tr>
</tbody></table><br></div><div id="gmail-m_1877032139338450650x6e34d26becf445d4ad84c1af0aceb2c0">This table with observed values may make a difference with reference to the expected values. The latter are determined using the margin totals:</div><div id="gmail-m_1877032139338450650x6e34d26becf445d4ad84c1af0aceb2c0"><br></div><div id="gmail-m_1877032139338450650x6e34d26becf445d4ad84c1af0aceb2c0"><div id="gmail-m_1877032139338450650xa0ff612093ea4dc48236761251dfe5c1">
<table style="width:218pt" width="292" cellspacing="0" cellpadding="0" border="0">
<colgroup><col style="width:47pt" width="63">
<col style="width:61pt" width="82">
<col style="width:63pt" width="84">
<col style="width:47pt" width="63">
</colgroup><tbody><tr style="height:14pt" height="19">
<td class="gmail-m_1877032139338450650xl65" style="height:14pt;width:47pt" width="63" height="19">Expected</td>
<td class="gmail-m_1877032139338450650xl65" style="border-left:medium none;width:61pt" width="82">Treatment </td>
<td class="gmail-m_1877032139338450650xl65" style="border-left:medium none;width:63pt" width="84">No treatment</td>
<td class="gmail-m_1877032139338450650xl66" style="width:47pt" width="63"></td>
</tr>
<tr style="height:14pt" height="19">
<td class="gmail-m_1877032139338450650xl65" style="height:14pt;border-top:medium none" height="19">male</td>
<td class="gmail-m_1877032139338450650xl65" style="border-top:medium none;border-left:medium none" align="right">140</td>
<td class="gmail-m_1877032139338450650xl65" style="border-top:medium none;border-left:medium none" align="right">79</td>
<td class="gmail-m_1877032139338450650xl66" align="right">219</td>
</tr>
<tr style="height:14pt" height="19">
<td class="gmail-m_1877032139338450650xl65" style="height:14pt;border-top:medium none" height="19">female</td>
<td class="gmail-m_1877032139338450650xl65" style="border-top:medium none;border-left:medium none" align="right">102</td>
<td class="gmail-m_1877032139338450650xl65" style="border-top:medium none;border-left:medium none" align="right">58</td>
<td class="gmail-m_1877032139338450650xl66" align="right">160</td>
</tr>
<tr style="height:14pt" height="19">
<td class="gmail-m_1877032139338450650xl66" style="height:14pt" height="19"></td>
<td class="gmail-m_1877032139338450650xl66" align="right">242</td>
<td class="gmail-m_1877032139338450650xl66" align="right">137</td>
<td class="gmail-m_1877032139338450650xl66" align="right">379</td>
</tr>
</tbody></table><br></div><div id="gmail-m_1877032139338450650xa0ff612093ea4dc48236761251dfe5c1">for example for the first cell: 219 * 242/ 379 = 140.</div><div id="gmail-m_1877032139338450650xa0ff612093ea4dc48236761251dfe5c1"><br></div><div id="gmail-m_1877032139338450650xa0ff612093ea4dc48236761251dfe5c1">Each corresponding cell contributes to the chi-square with the so-called standardized residual to the chi-square: </div><div id="gmail-m_1877032139338450650xa0ff612093ea4dc48236761251dfe5c1"><br></div><div id="gmail-m_1877032139338450650xa0ff612093ea4dc48236761251dfe5c1">
<div>
<img src="cid:1731f848f032c91aec1" width="213" height="60">
</div><div>The residual of the chi-square is the square root of this:</div><div><br></div><div>
<div> </div>
<div>
<img src="cid:1731f848f042d7326d2" width="211" height="64">
</div>
<div> This is a <i>z-</i>statistic. For the four cells of the matrix, we can derive:</div><div><div id="gmail-m_1877032139338450650xca030115c06040e"><div id="gmail-m_1877032139338450650x6e34d26becf445d4ad84c1af0aceb2c0"><div id="gmail-m_1877032139338450650xa0ff612093ea4dc48236761251dfe5c1"><div><div></div><div></div></div></div></div></div></div><div id="gmail-m_1877032139338450650x092434b26b0d4b3e840a7b5fd0361586">
<table style="width:171pt" width="229" cellspacing="0" cellpadding="0" border="0">
<colgroup><col style="width:47pt" width="63">
<col style="width:61pt" width="82">
<col style="width:63pt" width="84">
</colgroup><tbody><tr style="height:14pt" height="19">
<td style="height:14pt;width:47pt" width="63" height="19"></td>
<td style="width:61pt" width="82">Treatment</td>
<td style="width:63pt" width="84">No Treatment</td>
</tr>
<tr style="height:14pt" height="19">
<td class="gmail-m_1877032139338450650xl65" style="height:14pt" height="19">male</td>
<td class="gmail-m_1877032139338450650xl66" style="border-left:medium none" align="right">-1.37</td>
<td class="gmail-m_1877032139338450650xl66" style="border-left:medium none" align="right">1.82</td>
</tr>
<tr style="height:14pt" height="19">
<td class="gmail-m_1877032139338450650xl65" style="height:14pt;border-top:medium none" height="19">female</td>
<td class="gmail-m_1877032139338450650xl66" style="border-top:medium none;border-left:medium none" align="right">1.60</td>
<td class="gmail-m_1877032139338450650xl67" style="border-top:medium none;border-left:medium none" align="right">-2.13</td>
</tr>
</tbody></table></div><div> </div></div><div>The difference between expected and observed is only significant (p <.05) for women without treatment:<i> z </i>= - 2.13. The threshold is 1.96 for the 5% level. The z-values can directly be used for the measurement of Bateson's information : "a difference that makes a difference. The difference of receiving treatment makes a difference for all four categories, but this difference is in tis case only significant for women without treatment. </div><div><br></div><div>Thus, Bateson-type information can be measured. There is also a link to Shannon-type information via the log-likelyhood chi-square (which is a Shannon-type measure.)</div><div><br></div><div>Problems solved? One can measure the value of the difference that makes a difference.</div><div><br></div><div>Best,</div><div>Loet</div><div><br></div></div></div></div></div>_______________________________________________<br>
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