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body { font-family: Segoe UI; font-size: 12pt; } </style></head><body><div id="xca030115c06040e">Dear Marcus and colleagues: </div><div id="xca030115c06040e"><br /></div><div id="xca030115c06040e"><br /></div><div id="xca030115c06040e"><blockquote cite="CACJqm9zq_JtFFeCM2NxDRP0N1eT3nZmdMP+h3Y9JW8xDEbVmwQ@mail.gmail.com" type="cite" class="cite2"><div dir="ltr"><div dir="ltr"><div dir="ltr"><div><div dir="ltr" class="gmail_signature"><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="xca030115c06040e"><br /></div><div id="xca030115c06040e">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="xca030115c06040e"><br /><div id="x6e34d26becf445d4ad84c1af0aceb2c0">
<table border="0" cellpadding="0" cellspacing="0" width="292" style="width: 218pt;" xmlns="http://www.w3.org/TR/REC-html40">
<colgroup><col width="63" style="width:47pt" />
<col width="82" style="mso-width-source:userset;mso-width-alt:2918;width:61pt" />
<col width="84" style="mso-width-source:userset;mso-width-alt:2995;width:63pt" />
<col width="63" style="width:47pt" />
</colgroup><tbody><tr height="19" style="height:14.0pt">
<td height="19" class="xl65" width="63" style="height:14.0pt;width:47pt">Observed</td>
<td class="xl65" width="82" style="border-left:none;width:61pt">Treatment </td>
<td class="xl65" width="84" style="border-left:none;width:63pt">No treatment</td>
<td width="63" style="width:47pt"></td>
</tr>
<tr height="19" style="height:14.0pt">
<td height="19" class="xl65" style="height:14.0pt;border-top:none">male</td>
<td class="xl65" align="right" style="border-top:none;border-left:none">156</td>
<td class="xl65" align="right" style="border-top:none;border-left:none">63</td>
<td align="right">219</td>
</tr>
<tr height="19" style="height:14.0pt">
<td height="19" class="xl65" style="height:14.0pt;border-top:none">female</td>
<td class="xl65" align="right" style="border-top:none;border-left:none">86</td>
<td class="xl65" align="right" style="border-top:none;border-left:none">74</td>
<td align="right">160</td>
</tr>
<tr height="19" style="height:14.0pt">
<td height="19" style="height:14.0pt"></td>
<td align="right">242</td>
<td align="right">137</td>
<td align="right">379</td>
</tr>
</tbody></table><br /></div><div id="x6e34d26becf445d4ad84c1af0aceb2c0">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="x6e34d26becf445d4ad84c1af0aceb2c0"><br /></div><div id="x6e34d26becf445d4ad84c1af0aceb2c0"><div id="xa0ff612093ea4dc48236761251dfe5c1">
<table border="0" cellpadding="0" cellspacing="0" width="292" style="width: 218pt;" xmlns="http://www.w3.org/TR/REC-html40">
<colgroup><col width="63" style="width:47pt" />
<col width="82" style="mso-width-source:userset;mso-width-alt:2918;width:61pt" />
<col width="84" style="mso-width-source:userset;mso-width-alt:2995;width:63pt" />
<col width="63" style="width:47pt" />
</colgroup><tbody><tr height="19" style="height:14.0pt">
<td height="19" class="xl65" width="63" style="height:14.0pt;width:47pt">Expected</td>
<td class="xl65" width="82" style="border-left:none;width:61pt">Treatment </td>
<td class="xl65" width="84" style="border-left:none;width:63pt">No treatment</td>
<td class="xl66" width="63" style="width:47pt"></td>
</tr>
<tr height="19" style="height:14.0pt">
<td height="19" class="xl65" style="height:14.0pt;border-top:none">male</td>
<td class="xl65" align="right" style="border-top:none;border-left:none">140</td>
<td class="xl65" align="right" style="border-top:none;border-left:none">79</td>
<td class="xl66" align="right">219</td>
</tr>
<tr height="19" style="height:14.0pt">
<td height="19" class="xl65" style="height:14.0pt;border-top:none">female</td>
<td class="xl65" align="right" style="border-top:none;border-left:none">102</td>
<td class="xl65" align="right" style="border-top:none;border-left:none">58</td>
<td class="xl66" align="right">160</td>
</tr>
<tr height="19" style="height:14.0pt">
<td height="19" class="xl66" style="height:14.0pt"></td>
<td class="xl66" align="right">242</td>
<td class="xl66" align="right">137</td>
<td class="xl66" align="right">379</td>
</tr>
</tbody></table><br /></div><div id="xa0ff612093ea4dc48236761251dfe5c1">for example for the first cell: 219 * 242/ 379 = 140.</div><div id="xa0ff612093ea4dc48236761251dfe5c1"><br /></div><div id="xa0ff612093ea4dc48236761251dfe5c1">Each corresponding cell contributes to the chi-square with the so-called standardized residual to the chi-square: </div><div id="xa0ff612093ea4dc48236761251dfe5c1"><br /></div><div id="xa0ff612093ea4dc48236761251dfe5c1">
<div>
<img src="cid:em88e0d989-55ae-4a8b-a5f6-f946581f9f88@pc2014" 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:em210b5b42-2fea-4a4a-9b03-bbc8395bcb72@pc2014" 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="xca030115c06040e"><div id="x6e34d26becf445d4ad84c1af0aceb2c0"><div id="xa0ff612093ea4dc48236761251dfe5c1"><div><div></div><div></div></div></div></div></div></div><div id="x092434b26b0d4b3e840a7b5fd0361586">
<table border="0" cellpadding="0" cellspacing="0" width="229" style="width: 171pt;" xmlns="http://www.w3.org/TR/REC-html40">
<colgroup><col width="63" style="width:47pt" />
<col width="82" style="mso-width-source:userset;mso-width-alt:2918;width:61pt" />
<col width="84" style="mso-width-source:userset;mso-width-alt:2995;width:63pt" />
</colgroup><tbody><tr height="19" style="height:14.0pt">
<td height="19" width="63" style="height:14.0pt;width:47pt"></td>
<td width="82" style="width:61pt">Treatment</td>
<td width="84" style="width:63pt">No Treatment</td>
</tr>
<tr height="19" style="height:14.0pt">
<td height="19" class="xl65" style="height:14.0pt">male</td>
<td class="xl66" align="right" style="border-left:none">-1.37</td>
<td class="xl66" align="right" style="border-left:none">1.82</td>
</tr>
<tr height="19" style="height:14.0pt">
<td height="19" class="xl65" style="height:14.0pt;border-top:none">female</td>
<td class="xl66" align="right" style="border-top:none;border-left:none">1.60</td>
<td class="xl67" align="right" style="border-top:none;border-left:none">-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></body></html>