Calculate Confidence Intervals for Crosstabs with Numeric Questions

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


// Add confidence intervals to crosstabs showing Averages

// 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
// This version was current as of 2014-05-14
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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;
// Numeric results
if (table.availableStatistics.indexOf("Average") > -1) {
    var upper_statistic_to_replace = "Maximum";
    var lower_statistic_to_replace = "Minimum";
} else {
    form.ruleNotApplicable('it only applies to tables containing the Average in the Statistics - Cells');
}

if (table.availableStatistics.indexOf("Standard Error") == -1){
    table.suppressOutput('Confidence intervals not computed - the table should have the Average and Standard Error statistic available.');
    form.ruleNotApplicable('it only applies to tables where the Standard Error is available in the Statistics - Cells');
}

if (["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 Brown drop-down menu.');
    form.ruleNotApplicable('the table should have a Pick One or Pick Any question selected in the Brown drop-down menu.');
}

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

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]);

// Work out quantile of Student's T Distribution
var p_cutoff = p_level.getValue();


// Get relevant table stats
var means = table.get("Average");
var standard_errors = table.get("Standard Error");
var ns = table.get('Column n');
var upper_values = table.get(upper_statistic_to_replace);
var lower_values = table.get(lower_statistic_to_replace);


// Do the calculations
for (var row = 0; row < table.numberRows; row++) {
    for (var col = 0; col < table.numberColumns; col++) {
        var t_quantile = jStat.studentt.inv(p_cutoff / 2, ns[row][col]);
        var increment = t_quantile * standard_errors[row][col];
        lower_values[row][col] = means[row][col] - increment;
        upper_values[row][col] = means[row][col] + increment;
    }    
}

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

Notes:

  • This Rule applies the formula from the Default Confidence Intervals section of Confidence Interval which uses the quantiles of the Student's t-distribution. The rule uses the Average, Standard Error, and Column n statistics on your table.
  • Numbers may differ slightly (5th decimal place) from equivalent intervals computed on SUMMARY tables in Q due to different algorithms used to compute the quantiles of the t-Distribution.
  • This Rule uses the jStat JavaScript library to compute the quatiles of the Student's t-distribution.

See also