Advanced Estimation - Bivariate Statistics

To access this dialog:

Using the Advanced Estimation dialog, select Bivariate Statistics from the left-hand menu system.

This panel is used to calculate useful statistics regarding your input samples.

If no zonal control is being applied (i.e. no zones are selected on the Select Samples panel), you can generate statistics for the full sample set associated with your estimation scenario.
If zonal control is being applied, each zone or zone combination (if two zones have been defined) will appear as a button along the top of the panel. You can generate summary statistics for any zone or zone combination.
If custom zones have been defined, these will also appear as mutually-exclusive buttons. This lets you generate statistics relating to all zones or zone combinations assigned to the custom zone.

Statistics are shown in a 2-dimensional grid when Recalculate sample stats is selected. Samples are represented for the currently selected zone, custom zone or the full sample data, split into each variable/grade being considered for estimation.

Note: Advanced Estimation is part of the Studio RM toolset. Additional licensing modules aren't required.

For example, in the image below, a single zone field has been defined, containing absent values, 1 and 2 only. Two custom zones have been created. The first (All zones) contains all 3 zone values, including absent. The second custom zone contains only zones 1 and 2 (no absent zone data). A multivariate estimation is being setup (AU, CU). The statistic being reported is [Number of non-absent pairs]:

There are other statistics available, correlations, correlation p-values, covariances-ratios and paired T-test statistics. Calculating these can take some time, particularly with large sample inputs. You cancel the operation using <ESC>.

Field Details:

Bivariate Sample Statistics: this command group contains the following fields:

Recalculate sample stats: this button is enabled where a selection has been made for which a calculated statistic is not yet available. For example, if you introduce a new variable using the Select grade/variable fields list on the left, for which no existing value exists in memory.

Zone selection: a series of mutually-exclusive buttons that define the zone or zones for which statistics will be reported. This can include:

All Zones: a single button is displayed if no zones have been defined on the Select Samples panel. Statistics will be calculated across the full input sample data, or;
Unique Zone Values: if a single zone field has been defined, a button will appear for each unique zone attribute value that exists, or;
Unique Zone Combinations: if two zone fields have been defined, a button will appear for each unique combination of zone attribute values.

The above outcomes are mutually-exclusive. In addition to these buttons, custom zones will also appear if they have been defined beforehand. Selecting a custom zone button lets you generate statistics for multiple zones, potentially.

Statistic: what do you want to see? You can choose from any of the following options

Number of pairs: show the total number of sample pairs (which must have non-absent values) for each of the selected variable/grade field combinations.
- Colour Key – identify missing values
  
  Orange: zero pairs
  Blue: >0 pairs
Correlation Coefficient: a measure of the relationship between the nominated sample variables. The value of the coefficient lies between +1 and -1, where a positive value indicates a positive linear relationship (both variables increasing or both variables decreasing together) and a negative value indicates a negative linear relationship (one variable increasing as the other decreases). A coefficient close to +1 or -1 represents a very strong relationship and a coefficient close to zero a weak relationship (or in the case of zero, no relationship).
- Colour Key –see p-value below
Correlation P-value: the p-value is the probability that you would have found the current result if the correlation coefficient were in fact zero (the null hypothesis). For example, you are trying to determine if the relationship between gold and silver is significant; then we start with the ‘null hypothesis’ which, in this case is the statement that the gold and silver grade values are unrelated’. The p-value is a number between 0 and 1 representing the probability that this data could have arisen if the null hypothesis were true. The probability is shown as value between 0 (0%) and 1 (100%).
- Colour Key – this is based on both the correlation coefficient (cc) and p-value (p)
  
  Grey: on main diagonal
  Blue: if cc ≤ 0.3 or  p > 0.05
  Orange:if  0.3 ≤ cc ≤ 0.6  and  p ≤ 0.05
  Green:cc > 0.6  and  p ≤ 0.05
Covariance: the covariance provides a measure of how much two random variables change together.If the greater values of one variable mainly correspond with the greater values of the other variable, and the same holds for the lesser values, i.e., the variables tend to show similar behavior, the covariance is positive. In the opposite case, when the greater values of one variable mainly correspond to the lesser values of the other, i.e., the variables tend to show opposite behavior, the covariance is negative. The sign of the covariance therefore shows the tendency in the linear relationship between the variables.

The magnitude of the covariance is not easy to interpret. Thenormalized version of the covariance, thecorrelation coefficient, and the p-value, shows by their magnitude the strength of the linear relationship.

The covariance of a grade with itself is the variance of the grade.
- Colour Key – see p-value above, except that it is blue on the main diagonal
F-Ratio:the F Ratio test can be used to help determine whether the variances of two sets of measured values with different numbers of samples are significantly different from each other. For example if a set of samples are analyzed by two laboratories and you want to test whether there is a difference between them. The value shown in the matrix should be compared to a set of standard values available from statistics text books or from the web: http://www.itl.nist.gov/div898/handbook/eda/section3/eda3673.htm
- Colour Key – main diagonal grey, otherwise blue
Paired T-Test: the paired t test can be used to help determine whether the means of two paired sets of measured values are significantly different from each other. For example if a set of samples are analyzed by two laboratories and you want to test whether there is a difference between them. For further information see:
http://www.itl.nist.gov/div898/handbook/eda/section3/eda53.htm.
- Colour Key
  Grey: on main diagonal
  Blue: if p ≤ 0.05
  Orange:if p > 0.05

The lookup table: once you have defined the scope and type of statistical information you want to view, the color-coded correlation table below will automatically update to indicate the relationship between variables for that particular context.

Example - Custom Zone Statistics

In this example, an input sample file (holes_eg1) is chosen. It contains 3 variables to be estimated, and multiple zones contained within a ROCK attribute. Estimation will be restricted to a single zone, but using samples from multiple zones. As such, a 'soft boundary' estimation is being performed.

A new scenario is created using the Scenario Setup panel.

The input sample file is selected. Three grade fields: AG, AU and CU have been selected and the Zone 1 field is ”ROCK”. This is done using the Select Samples panel:

ROCK is the domain field for statistical and variogram analysis. Coordinate fields have been assigned automatically as have the fields BHID, FROM and TO.

ROCK contains 4 unique values: 0, 1, 2 and 3.

In this example, soft boundary analysis is desired; estimation will be performed within ROCK zone 1 but with estimates influenced by samples from both zones 1 and 2. This example aims to find out the number of non-absent sample pairs that will contribute to the multivariate estimation.

To generate statistics for multiples zones, a custom zone is required. This is set up using the Define Custom Zones panel, e.g.:

Now that a custom zone has been defined, it appears on the Bivariate Statistics panel, alongside the unique zone values for ROCK:

Selecting [1+2] and the [Number of non-absent pairs] Statistic means that generated statistics will represents both ROCK zones 1 and 2. Data is shown for each grade/variable combination, e.g.:

Other statistics are also available, e.g.:

All three grade combinations have a p-value of less than 0.001. This means that there is less than 0.1% probability that this data would be created if the null hypothesis (no correlation) were true. In this example all three correlations are strong.

The correlation coefficient of 0.63 between AU and AG is the highest of the three grade combinations. This is shown in green as the value is above 0.6 and the p-value <0.05. The correlation coefficients for the other two lie between 0.3 and 0.6 and p-values are <0.05 so they are coloured orange.

The relationship between grades can be displayed graphically using the Point/Line Plot option on the Sample Analysis ribbon. The graphic below shows the relationship between AU, AG and CU for ROCK 1.

The correlation coefficients and the correlation p-values are shown in the Sample Statistics matrix on the Advanced Estimation dialog. It can be seen that the p-values are all zero showing that for ROCK 1 there is an extremely small probability that there is no correlation between any of the grades.

The corresponding scatterplots for ROCK 3 are shown below, and also the correlation coefficients and correlation p-values. The correlation coefficient for CU/AG (0.18) is coloured orange because the correlation p-value Is >0.05 (0.00519). This shows that the CU/AG correlation is weaker for ROCK 3 than for ROCK 1.

	Related Topics
	Advanced Estimation Introduction Scenario Setup Select Samples Define Zones Define Estimations