Standardize binomial variable for non-constant meta-population - binomial z-scores












0












$begingroup$


Let's say I'm doing a meta-analysis of a general experimental protocol that has been applied to some experiments (with 1,0 type outcomes) across a variety of experiment-type sub-groups. I want to test the hypothesis that experiments using the protocol are more likely to achieve their endpoint (x=1) than do other experiments of the same type sub-group that do not utilize the protocol. There may be only one or two experiments that utilized the protocol in each sub-group. All the experiments across all sub-groups are basically similar, except that the probability of success (p) can vary a fair amount across sub-groups (though p is constant within a sub-group). My instinct is to standardize each of the instances of the protocol-experiment outcomes (x) by calculating a Bernoulli z-score for each as z = (x - p) / sqrt{ p (1 - p) }, where p refers to the measured success-ratio across the entire sub-group that the (protocol-using) experiment x was drawn from. Then using this collection of standardized z values, what would be the test for my hypothesis? Would the (squared) standardized values follow a chi-squared distribution, as per the K-proportions theorem? Let me say, the populations of each sub-group are large (>3000), but the number of subgroups / protocol-using experiments is relatively small (10-20, depending), which means I'd be using small degrees of freedom despite having standardized on large sets ... ? Am I at all on the right track? Thanks for your help.










share|cite|improve this question











$endgroup$

















    0












    $begingroup$


    Let's say I'm doing a meta-analysis of a general experimental protocol that has been applied to some experiments (with 1,0 type outcomes) across a variety of experiment-type sub-groups. I want to test the hypothesis that experiments using the protocol are more likely to achieve their endpoint (x=1) than do other experiments of the same type sub-group that do not utilize the protocol. There may be only one or two experiments that utilized the protocol in each sub-group. All the experiments across all sub-groups are basically similar, except that the probability of success (p) can vary a fair amount across sub-groups (though p is constant within a sub-group). My instinct is to standardize each of the instances of the protocol-experiment outcomes (x) by calculating a Bernoulli z-score for each as z = (x - p) / sqrt{ p (1 - p) }, where p refers to the measured success-ratio across the entire sub-group that the (protocol-using) experiment x was drawn from. Then using this collection of standardized z values, what would be the test for my hypothesis? Would the (squared) standardized values follow a chi-squared distribution, as per the K-proportions theorem? Let me say, the populations of each sub-group are large (>3000), but the number of subgroups / protocol-using experiments is relatively small (10-20, depending), which means I'd be using small degrees of freedom despite having standardized on large sets ... ? Am I at all on the right track? Thanks for your help.










    share|cite|improve this question











    $endgroup$















      0












      0








      0





      $begingroup$


      Let's say I'm doing a meta-analysis of a general experimental protocol that has been applied to some experiments (with 1,0 type outcomes) across a variety of experiment-type sub-groups. I want to test the hypothesis that experiments using the protocol are more likely to achieve their endpoint (x=1) than do other experiments of the same type sub-group that do not utilize the protocol. There may be only one or two experiments that utilized the protocol in each sub-group. All the experiments across all sub-groups are basically similar, except that the probability of success (p) can vary a fair amount across sub-groups (though p is constant within a sub-group). My instinct is to standardize each of the instances of the protocol-experiment outcomes (x) by calculating a Bernoulli z-score for each as z = (x - p) / sqrt{ p (1 - p) }, where p refers to the measured success-ratio across the entire sub-group that the (protocol-using) experiment x was drawn from. Then using this collection of standardized z values, what would be the test for my hypothesis? Would the (squared) standardized values follow a chi-squared distribution, as per the K-proportions theorem? Let me say, the populations of each sub-group are large (>3000), but the number of subgroups / protocol-using experiments is relatively small (10-20, depending), which means I'd be using small degrees of freedom despite having standardized on large sets ... ? Am I at all on the right track? Thanks for your help.










      share|cite|improve this question











      $endgroup$




      Let's say I'm doing a meta-analysis of a general experimental protocol that has been applied to some experiments (with 1,0 type outcomes) across a variety of experiment-type sub-groups. I want to test the hypothesis that experiments using the protocol are more likely to achieve their endpoint (x=1) than do other experiments of the same type sub-group that do not utilize the protocol. There may be only one or two experiments that utilized the protocol in each sub-group. All the experiments across all sub-groups are basically similar, except that the probability of success (p) can vary a fair amount across sub-groups (though p is constant within a sub-group). My instinct is to standardize each of the instances of the protocol-experiment outcomes (x) by calculating a Bernoulli z-score for each as z = (x - p) / sqrt{ p (1 - p) }, where p refers to the measured success-ratio across the entire sub-group that the (protocol-using) experiment x was drawn from. Then using this collection of standardized z values, what would be the test for my hypothesis? Would the (squared) standardized values follow a chi-squared distribution, as per the K-proportions theorem? Let me say, the populations of each sub-group are large (>3000), but the number of subgroups / protocol-using experiments is relatively small (10-20, depending), which means I'd be using small degrees of freedom despite having standardized on large sets ... ? Am I at all on the right track? Thanks for your help.







      probability-theory statistical-inference binomial-distribution hypothesis-testing






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited Jan 2 at 14:00







      fvc

















      asked Jan 1 at 16:34









      fvcfvc

      43




      43






















          0






          active

          oldest

          votes











          Your Answer





          StackExchange.ifUsing("editor", function () {
          return StackExchange.using("mathjaxEditing", function () {
          StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
          StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
          });
          });
          }, "mathjax-editing");

          StackExchange.ready(function() {
          var channelOptions = {
          tags: "".split(" "),
          id: "69"
          };
          initTagRenderer("".split(" "), "".split(" "), channelOptions);

          StackExchange.using("externalEditor", function() {
          // Have to fire editor after snippets, if snippets enabled
          if (StackExchange.settings.snippets.snippetsEnabled) {
          StackExchange.using("snippets", function() {
          createEditor();
          });
          }
          else {
          createEditor();
          }
          });

          function createEditor() {
          StackExchange.prepareEditor({
          heartbeatType: 'answer',
          autoActivateHeartbeat: false,
          convertImagesToLinks: true,
          noModals: true,
          showLowRepImageUploadWarning: true,
          reputationToPostImages: 10,
          bindNavPrevention: true,
          postfix: "",
          imageUploader: {
          brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
          contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
          allowUrls: true
          },
          noCode: true, onDemand: true,
          discardSelector: ".discard-answer"
          ,immediatelyShowMarkdownHelp:true
          });


          }
          });














          draft saved

          draft discarded


















          StackExchange.ready(
          function () {
          StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3058627%2fstandardize-binomial-variable-for-non-constant-meta-population-binomial-z-scor%23new-answer', 'question_page');
          }
          );

          Post as a guest















          Required, but never shown

























          0






          active

          oldest

          votes








          0






          active

          oldest

          votes









          active

          oldest

          votes






          active

          oldest

          votes
















          draft saved

          draft discarded




















































          Thanks for contributing an answer to Mathematics Stack Exchange!


          • Please be sure to answer the question. Provide details and share your research!

          But avoid



          • Asking for help, clarification, or responding to other answers.

          • Making statements based on opinion; back them up with references or personal experience.


          Use MathJax to format equations. MathJax reference.


          To learn more, see our tips on writing great answers.




          draft saved


          draft discarded














          StackExchange.ready(
          function () {
          StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3058627%2fstandardize-binomial-variable-for-non-constant-meta-population-binomial-z-scor%23new-answer', 'question_page');
          }
          );

          Post as a guest















          Required, but never shown





















































          Required, but never shown














          Required, but never shown












          Required, but never shown







          Required, but never shown

































          Required, but never shown














          Required, but never shown












          Required, but never shown







          Required, but never shown







          Popular posts from this blog

          Ellipse (mathématiques)

          Quarter-circle Tiles

          Mont Emei