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For anyone else who comes across the same problem, this worked for me:

// include the library code:
#include <SPI.h>
// Set Constants
const int adcChipSelectPin = 8;      // set pin 8 as the chip select for the ADC:
// Start setup function:
void setup() {
pinMode (adcChipSelectPin, OUTPUT);
// set the ChipSelectPins high initially:
digitalWrite(adcChipSelectPin, HIGH);

// initialise SPI:
SPI.begin();
SPI.setBitOrder(MSBFIRST);         // Not strictly needed but just to be sure.
SPI.setDataMode(SPI_MODE0);        // Not strictly needed but just to be sure.
Serial.begin(9600);

//Clock Speed: Master clock/divider
//84Mhz/64 =  1.3 MHz
SPI.setClockDivider(SPI_CLOCK_DIV64);

//Send dummy values over when ADC is turned on 
//Hold Din High for 16 clock cycles twice 
digitalWrite(adcChipSelectPin, LOW);
SPI.transfer(0xFF);
SPI.transfer(0xFF);
digitalWrite(adcChipSelectPin, HIGH);

digitalWrite(adcChipSelectPin, LOW);
SPI.transfer(0xFF);
SPI.transfer(0xFF);
digitalWrite(adcChipSelectPin, HIGH);
asm volatile("nop"); 
asm volatile("nop"); 
asm volatile("nop"); 
asm volatile("nop"); 
asm volatile("nop"); 
//Send dummy values, next read should have correct values

digitalWrite(adcChipSelectPin, LOW);
byte FirstByte = 0x83;
byte SecondByte = 0x50;
byte response1 = SPI.transfer(FirstByte); 
byte response2 = SPI.transfer(SecondByte); 

digitalWrite(adcChipSelectPin, HIGH);

asm volatile("nop"); 
asm volatile("nop"); 
asm volatile("nop"); 
asm volatile("nop"); 
asm volatile("nop"); 
///////////////////////////////////////////////////////////////////////////////////

//50ns quiet time?
asm volatile("nop"); 
asm volatile("nop"); 
asm volatile("nop"); 
asm volatile("nop"); 
asm volatile("nop"); 



} // End setup function.

// Start loop function:
void loop() {
 float voltage[2];
 for (int i = 1; i < 17; i++) {
 voltage[i] = readAdc(0, 1);
 //50ns quiet time? 
 asm volatile("nop"); 
 asm volatile("nop"); 
 asm volatile("nop"); 
 asm volatile("nop"); 
 asm volatile("nop"); 


 }
 }// End of loop function.
 //Function to read the ADC, accepts the channel to be read.
 float readAdc(int channel, int slave) {

 float value;
 if (slave == 0) { 

  byte FirstByte = 0x83;
  byte SecondByte = 0x50;
  FirstByte |= channel << 2;
  noInterrupts(); // disable interupts to prepare to send address data to the ADC.
digitalWrite(adcChipSelectPin, LOW); // take the Chip Select pin low to select the ADC.
//Write = 1
//SEQ = 0
// ADD3-ADD0 = 0000
//PM1 = PM0 = 1
//Shadow = 0
//WEAK/Tri = 1
//Range = 0
//Coding = 1
//0b10000010 1010 0000
//extra 4 zeros to fill
 
//May need to wait 16 * (1/freq)
//16 * (1/1.3MHz) 
byte response1 = SPI.transfer(FirstByte); 
byte response2 = SPI.transfer(SecondByte); 
digitalWrite(adcChipSelectPin, HIGH);     
    
//digitalWrite(adcChipSelectPin, LOW); 
//byte response1 = SPI.transfer(0x00); 
//byte response2 = SPI.transfer(0x00);
//SPI.transfer(FirstByte);
//byte response1 = SPI.transfer(SecondByte);
//byte response2 = SPI.transfer(0x00)

//digitalWrite(adcChipSelectPin, HIGH); // take the Chip Select pin high to de-select the ADC.
interrupts(); // Enable interupts.
//Serial.println(response1);
Serial.println(response2);
byte channel = (response1 >> 4);
//byte value = (response1 << 4);
//byte digitalValue = (response1 << 4) | response2; 
//Serial.println(digitalValue);
//Serial.println(channel);
byte digitalValue = 0x00;
float value = (float(digitalValue) * 3.3) / 4096.000; // The digital value is 
converted to an analogue voltage using a VREF of 2.048V.
//Serial.print(channel);
//Serial.print("\t");
//Serial.println(value);
}