Calculate Confidence Intervals for Crosstabs with Categorical Questions

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This example may be used to compute confidence intervals for crosstabs where there is a categorical question (Pick One or Pick Any) selected in the Blue and Brown drop-down menu. As the rule borrows statistics from the table (by default it takes over the Index and Missing n statistics), you should rename these statistics by right-clicking your table, selecting Table Options, and changing the names of Index and Missing n to Upper Confidence Interval and Lower Confidence Interval in the Override Text column of the Output Text column.

To use this snippet:

  1. Select your table.
  2. Select Automate > Custom Rule.
  3. Paste in the code from below.
  4. (Optional) change the statistics used in upper_statistic_to_replace and lower_statistic_to_replace.
  5. Click the 'play' icon and close.
  6. Choose your Significance level. By default this is 0.05 to match Q's default Overall Significance Level setting, but if you are testing at a different level then change this number.
  7. Right-click on your table and select Statistics - Cells > Index and Missing n.
  8. Right-click your table, select Table Options, click Output Text, and change the names of Index and Missing n to Upper Confidence Interval and Lower Confidence Interval in the column of the Override Text column.


// Add confidence intervals to crosstabs

// Use jstat for computing t distribution quantiles
// Source: https://github.com/jstat/jstat
// Documentation: https://github.com/jstat/jstat/blob/master/doc/md/distributions.md
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h=0,i=c.length,j=[],k=[],l=[],m,n,o,p,q;for(;h<i;h++){j[h]=[],k[h]=[],l[h]=[];for(m=0;m<i;m++)h>m?(j[h][m]=c[h][m],k[h][m]=l[h][m]=0):h<m?(k[h][m]=c[h][m],j[h][m]=l[h][m]=0):(l[h][m]=c[h][m],j[h][m]=k[h][m]=0)}p=a.multiply(a.inv(a.add(l,a.multiply(j,g))),a.subtract(a.multiply(l,1-g),a.multiply(k,g))),o=a.multiply(a.multiply(a.inv(a.add(l,a.multiply(j,g))),d),g),n=e,q=a.add(a.multiply(p,e),o),h=2;while(b.abs(a.norm(a.subtract(q,n)))>f)n=q,q=a.add(a.multiply(p,n),o),h++;return q},householder:function(c){var d=c.length,e=c[0].length,f=0,g=[],h=[],i,j,k,l,m;for(;f<d-1;f++){i=0;for(l=f+1;l<e;l++)i+=c[l][f]*c[l][f];m=c[f+1][f]>0?-1:1,i=m*b.sqrt(i),j=b.sqrt((i*i-c[f+1][f]*i)/2),g=a.zeros(d,1),g[f+1][0]=(c[f+1][f]-i)/(2*j);for(k=f+2;k<d;k++)g[k][0]=c[k][f]/(2*j);h=a.subtract(a.identity(d,e),a.multiply(a.multiply(g,a.transpose(g)),2)),c=a.multiply(h,a.multiply(c,h))}return c},QR:function(c,d){var e=c.length,f=c[0].length,g=0,h=[],i=[],j=[],k,l,m,n,o,p;for(;g<e-1;g++){l=0;for(k=g+1;k<f;k++)l+=c[k][g]*c[k][g];o=c[g+1][g]>0?-1:1,l=o*b.sqrt(l),m=b.sqrt((l*l-c[g+1][g]*l)/2),h=a.zeros(e,1),h[g+1][0]=(c[g+1][g]-l)/(2*m);for(n=g+2;n<e;n++)h[n][0]=c[n][g]/(2*m);i=a.subtract(a.identity(e,f),a.multiply(a.multiply(h,a.transpose(h)),2)),c=a.multiply(i,c),d=a.multiply(i,d)}for(g=e-1;g>=0;g--){p=0;for(k=g+1;k<=f-1;k++)p=j[k]*c[g][k];j[g]=d[g][0]/c[g][g]}return j},jacobi:function(c){var d=1,e=0,f=c.length,g=a.identity(f,f),h=[],i,j,k,l,m,n,o,p;while(d===1){e++,n=c[0][1],l=0,m=1;for(j=0;j<f;j++)for(k=0;k<f;k++)j!=k&&n<b.abs(c[j][k])&&(n=b.abs(c[j][k]),l=j,m=k);c[l][l]===c[m][m]?o=c[l][m]>0?b.PI/4:-b.PI/4:o=b.atan(2*c[l][m]/(c[l][l]-c[m][m]))/2,p=a.identity(f,f),p[l][l]=b.cos(o),p[l][m]=-b.sin(o),p[m][l]=b.sin(o),p[m][m]=b.cos(o),g=a.multiply(g,p),i=a.multiply(a.multiply(a.inv(p),c),p),c=i,d=0;for(j=1;j<f;j++)for(k=1;k<f;k++)j!=k&&b.abs(c[j][k])>.001&&(d=1)}for(j=0;j<f;j++)h.push(c[j][j]);return[g,h]},rungekutta:function(a,b,c,d,e,f){var g,h,i,j,k;if(f===2)while(d<=c)g=b*a(d,e),h=b*a(d+b,e+g),i=e+(g+h)/2,e=i,d+=b;if(f===4)while(d<=c)g=b*a(d,e),h=b*a(d+b/2,e+g/2),j=b*a(d+b/2,e+h/2),k=b*a(d+b,e+j),i=e+(g+2*h+2*j+k)/6,e=i,d+=b;return e},romberg:function(a,c,d,e){var f=0,g=(d-c)/2,h=[],i=[],j=[],k,l,m,n,o,p;while(f<e/2){o=a(c);for(m=c,n=0;m<=d;m+=g,n++)h[n]=m;k=h.length;for(m=1;m<k-1;m++)o+=(m%2!==0?4:2)*a(h[m]);o=g/3*(o+a(d)),j[f]=o,g/=2,f++}l=j.length,k=1;while(l!==1){for(m=0;m<l-1;m++)i[m]=(b.pow(4,k)*j[m+1]-j[m])/(b.pow(4,k)-1);l=i.length,j=i,i=[],k++}return j},richardson:function(a,c,d,e){function f(a,b){var c=0,d=a.length,e;for(;c<d;c++)a[c]===b&&(e=c);return e}var g=a.length,h=b.abs(d-a[f(a,d)+1]),i=0,j=[],k=[],l,m,n,o,p;while(e>=h)l=f(a,d+e),m=f(a,d),j[i]=(c[l]-2*c[m]+c[2*m-l])/(e*e),e/=2,i++;o=j.length,n=1;while(o!=1){for(p=0;p<o-1;p++)k[p]=(b.pow(4,n)*j[p+1]-j[p])/(b.pow(4,n)-1);o=k.length,j=k,k=[],n++}return j},simpson:function(a,b,c,d){var e=(c-b)/d,f=a(b),g=[],h=b,i=0,j=1,k;for(;h<=c;h+=e,i++)g[i]=h;k=g.length;for(;j<k-1;j++)f+=(j%2!==0?4:2)*a(g[j]);return e/3*(f+a(c))},hermite:function(a,b,c,d){var e=a.length,f=0,g=0,h=[],i=[],j=[],k=[],l;for(;g<e;g++){h[g]=1;for(l=0;l<e;l++)g!=l&&(h[g]*=(d-a[l])/(a[g]-a[l]));i[g]=0;for(l=0;l<e;l++)g!=l&&(i[g]+=1/(a[g]-a[l]));j[g]=(1-2*(d-a[g])*i[g])*h[g]*h[g],k[g]=(d-a[g])*h[g]*h[g],f+=j[g]*b[g]+k[g]*c[g]}return f},lagrange:function(a,b,c){var d=0,e=0,f,g,h=a.length;for(;e<h;e++){g=b[e];for(f=0;f<h;f++)e!=f&&(g*=(c-a[f])/(a[e]-a[f]));d+=g}return d},cubic_spline:function(b,c,d){var e=b.length,f=0,g,h=[],i=[],j=[],k=[],l=[],m=[],n=[];for(;f<e-1;f++)l[f]=b[f+1]-b[f];j[0]=0;for(f=1;f<e-1;f++)j[f]=3/l[f]*(c[f+1]-c[f])-3/l[f-1]*(c[f]-c[f-1]);for(f=1;f<e-1;f++)h[f]=[],i[f]=[],h[f][f-1]=l[f-1],h[f][f]=2*(l[f-1]+l[f]),h[f][f+1]=l[f],i[f][0]=j[f];k=a.multiply(a.inv(h),i);for(g=0;g<e-1;g++)m[g]=(c[g+1]-c[g])/l[g]-l[g]*(k[g+1][0]+2*k[g][0])/3,n[g]=(k[g+1][0]-k[g][0])/(3*l[g]);for(g=0;g<e;g++)if(b[g]>d)break;return g-=1,c[g]+(d-b[g])*m[g]+a.sq(d-b[g])*k[g]+(d-b[g])*a.sq(d-b[g])*n[g]},gauss_quadrature:function(){},PCA:function(b){var c=b.length,d=b[0].length,e=!1,f=0,g,h,i=[],j=[],k=[],l=[],m=[],n=[],o=[],p=[],q=[],r=[];for(f=0;f<c;f++)i[f]=a.sum(b[f])/d;for(f=0;f<d;f++){o[f]=[];for(g=0;g<c;g++)o[f][g]=b[g][f]-i[g]}o=a.transpose(o);for(f=0;f<c;f++){p[f]=[];for(g=0;g<c;g++)p[f][g]=a.dot([o[f]],[o[g]])/(d-1)}k=a.jacobi(p),q=k[0],j=k[1],r=a.transpose(q);for(f=0;f<j.length;f++)for(g=f;g<j.length;g++)j[f]<j[g]&&(h=j[f],j[f]=j[g],j[g]=h,l=r[f],r[f]=r[g],r[g]=l);n=a.transpose(o);for(f=0;f<c;f++){m[f]=[];for(g=0;g<n.length;g++)m[f][g]=a.dot([r[f]],[n[g]])}return[b,j,r,m]}}),function(b){for(var c=0;c<b.length;c++)(function(b){a.fn[b]=function(c,d){var e=this;return d?(setTimeout(function(){d.call(e,a.fn[b].call(e,c))},15),this):a(a[b](this,c))}})(b[c])}("add divide multiply subtract dot pow abs norm angle".split(" "))}(this.jStat,Math),function(a,b){var c=[].slice,d=a.utils.isNumber;a.extend({zscore:function(){var b=c.call(arguments);return d(b[1])?(b[0]-b[1])/b[2]:(b[0]-a.mean(b[1]))/a.stdev(b[1],b[2])},ztest:function(){var e=c.call(arguments);if(e.length===4){if(d(e[1])){var f=a.zscore(e[0],e[1],e[2]);return e[3]===1?a.normal.cdf(-b.abs(f),0,1):a.normal.cdf(-b.abs(f),0,1)*2}var f=e[0];return e[2]===1?a.normal.cdf(-b.abs(f),0,1):a.normal.cdf(-b.abs(f),0,1)*2}var f=a.zscore(e[0],e[1],e[3]);return e[1]===1?a.normal.cdf(-b.abs(f),0,1):a.normal.cdf(-b.abs(f),0,1)*2}}),a.extend(a.fn,{zscore:function(a,b){return(a-this.mean())/this.stdev(b)},ztest:function(c,d,e){var f=b.abs(this.zscore(c,e));return d===1?a.normal.cdf(-f,0,1):a.normal.cdf(-f,0,1)*2}}),a.extend({tscore:function(){var d=c.call(arguments);return d.length===4?(d[0]-d[1])/(d[2]/b.sqrt(d[3])):(d[0]-a.mean(d[1]))/(a.stdev(d[1],!0)/b.sqrt(d[1].length))},ttest:function(){var e=c.call(arguments),f;return e.length===5?(f=b.abs(a.tscore(e[0],e[1],e[2],e[3])),e[4]===1?a.studentt.cdf(-f,e[3]-1):a.studentt.cdf(-f,e[3]-1)*2):d(e[1])?(f=b.abs(e[0]),e[2]==1?a.studentt.cdf(-f,e[1]-1):a.studentt.cdf(-f,e[1]-1)*2):(f=b.abs(a.tscore(e[0],e[1])),e[2]==1?a.studentt.cdf(-f,e[1].length-1):a.studentt.cdf(-f,e[1].length-1)*2)}}),a.extend(a.fn,{tscore:function(a){return(a-this.mean())/(this.stdev(!0)/b.sqrt(this.cols()))},ttest:function(c,d){return d===1?1-a.studentt.cdf(b.abs(this.tscore(c)),this.cols()-1):a.studentt.cdf(-b.abs(this.tscore(c)),this.cols()-1)*2}}),a.extend({anovafscore:function(){var d=c.call(arguments),e,f,g,h,i,j,k,l;if(d.length===1){i=new Array(d[0].length);for(k=0;k<d[0].length;k++)i[k]=d[0][k];d=i}if(d.length===2)return a.variance(d[0])/a.variance(d[1]);f=new Array;for(k=0;k<d.length;k++)f=f.concat(d[k]);g=a.mean(f),e=0;for(k=0;k<d.length;k++)e+=d[k].length*b.pow(a.mean(d[k])-g,2);e/=d.length-1,j=0;for(k=0;k<d.length;k++){h=a.mean(d[k]);for(l=0;l<d[k].length;l++)j+=b.pow(d[k][l]-h,2)}return j/=f.length-d.length,e/j},anovaftest:function(){var b=c.call(arguments),e,f,g,h;if(d(b[0]))return 1-a.centralF.cdf(b[0],b[1],b[2]);anovafscore=a.anovafscore(b),e=b.length-1,g=0;for(h=0;h<b.length;h++)g+=b[h].length;return f=g-e-1,1-a.centralF.cdf(anovafscore,e,f)},ftest:function(b,c,d){return 1-a.centralF.cdf(b,c,d)}}),a.extend(a.fn,{anovafscore:function(){return a.anovafscore(this.toArray())},anovaftest:function(){var b=0,c;for(c=0;c<this.length;c++)b+=this[c].length;return a.ftest(this.anovafscore(),this.length-1,b-this.length)}}),a.extend({normalci:function(){var d=c.call(arguments),e=new Array(2),f;return d.length===4?f=b.abs(a.normal.inv(d[1]/2,0,1)*d[2]/b.sqrt(d[3])):f=b.abs(a.normal.inv(d[1]/2,0,1)*a.stdev(d[2])/b.sqrt(d[2].length)),e[0]=d[0]-f,e[1]=d[0]+f,e},tci:function(){var d=c.call(arguments),e=new Array(2),f;return d.length===4?f=b.abs(a.studentt.inv(d[1]/2,d[3]-1
)*d[2]/b.sqrt(d[3])):f=b.abs(a.studentt.inv(d[1]/2,d[2].length)*a.stdev(d[2],!0)/b.sqrt(d[2].length)),e[0]=d[0]-f,e[1]=d[0]+f,e},significant:function(a,b){return a<b}}),a.extend(a.fn,{normalci:function(b,c){return a.normalci(b,c,this.toArray())},tci:function(b,c){return a.tci(b,c,this.toArray())}})}(this.jStat,Math);



// Choose which stats to replace.
// NOTE: You need to rename these statistics using Edit > Table Options > Output Text and typing
// "Upper Confidence Interval" and "Lower Confidence Interval" for upper_statistic_to_replace and
// upper_statistic_to_replace, respectively.
var upper_statistic_to_replace;
var lower_statistic_to_replace;
// Categorical results
if (table.availableStatistics.indexOf("Column %") > -1) {
    var upper_statistic_to_replace = "Index";
    var lower_statistic_to_replace = "Missing n";
} else {
	table.suppressOutput('Confidence intervals not computed - it only applies to tables containing Column % in the Statistics - Cells');
    form.ruleNotApplicable('it only applies to tables containing Column % in the Statistics - Cells');
}

if (["Pick One", "Pick Any"].indexOf(table.blueQuestion.questionType) == -1 || ["Pick One", "Pick Any"].indexOf(table.brownQuestion.questionType) == -1){
    table.suppressOutput('Confidence intervals not computed - the table should have a Pick One or Pick Any question selected in the Blue and Brown drop-down menus.');
    form.ruleNotApplicable('the table should have a Pick One or Pick Any question selected in the Blue and Brown drop-down menus.');
}

// Set up controls for user input.
form.setSummary('Calculate confidence intervals on crosstabs with percentages');  
form.setHeading('Calculate confidence intervals on crosstabs with percentages');

var label = form.newLabel('Significance level:');
var p_level  = form.newNumericUpDown('Lower');
p_level.setIncrement(0.01);
p_level.setDefault(0.05);
p_level.setMaximum(1);
p_level.setMinimum(0);
form.setInputControls([label, p_level]);

var p_cutoff = p_level.getValue();

// Get relevant table stats
var n = table.get("n");
var col_n = table.get("Column n");
var col_percent = table.get("Column %");
var col_se = table.get("Column Standard Error");
var upper_values = table.get(upper_statistic_to_replace);
var lower_values = table.get(lower_statistic_to_replace);
var isWeight = table.weight != null;

// Do the calculations
for (var row = 0; row < table.numberRows; row++) {
    for (var col = 0; col < table.numberColumns; col++) {
        var z = jStat.normal.inv(p_cutoff/2,0,1);

        if (isWeight) {
            var increment = z*col_se[row][col]*100;
            lower_values[row][col] = col_percent[row][col] + increment;
            upper_values[row][col] = col_percent[row][col] - increment;
        } else {    
            var calc1 = col_n[row][col] + Math.pow(z,2);
            var calc2 = (n[row][col] + Math.pow(z,2)/2)/calc1;
            var calc3 = Math.sqrt(calc2*(1-calc2)/calc1);
            var increment = z*calc3;
            lower_values[row][col] = (calc2 + increment)*100;
            upper_values[row][col] = (calc2 - increment)*100;
        }

    }    
}

table.set(lower_statistic_to_replace, lower_values);
table.set(upper_statistic_to_replace, upper_values);

Notes:

  • This Rule applies one of two the formulas from the page Confidence Interval, depending on whether or not the table is weighted:
    • When the table is weighted, the rule applies the formula from the section called Default confidence intervals. This which uses the quantiles of the normal distribution.
    • When the table is not weighted, this rule applies the formula described in the section Confidence intervals for percentages with unweighted samples. This is the Agresti-Coull interval
  • The rule uses the Column %, Column Standard Error, n, and Column n statistics on your table.
  • This Rule uses the jStat JavaScript library to compute the quantiles of the normal distribution.

See also

Retrieved from "https://wiki2.q-researchsoftware.com/index.php?title=Calculate_Confidence_Intervals_for_Crosstabs_with_Categorical_Questions&oldid=37891"